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MATHEMATICS

Quarter 4 – Module 1

Illustrating The Measures of Position

for Ungrouped Data

Mathematics – Grade 10 Alternative Delivery Mode Quarter 4 – Module 1 : Illustrating the following Measures of Position for 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 Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Printed in the Philippines by ________________________ Department of Education – Region VII Schools Division of Negros Oriental Office Address: Kagawasan, Ave., Daro, Dumaguete City, Negros Oriental Tel #: (035) 225 2376 / 541 1117 E-mail Address: [email protected] Development Team of the Module Writer: Claudeth S. Mercado Editors: Maricel T. Tropezado, James D. Rodriguez, Maria Fatima M. Emperado, Shiela P. Repe Reviewers: Maricel T. Tropezado, James D. Rodriguez, Maria Fatima M. Emperado, Shiela P. Repe Layout Artist: Mylyn Q. Liboon, Lyka May S. Berlan Management Team: Senen Priscillo P. Paulin, CESO V Elisa L. Baguio, Ed.D. Joelyza M. Arcilla, Ed.D., CESE Rosela R. Abiera Marcelo K. Palispis, JD, Ed.D Maricel S. Rasid Nilita L. Ragay, Ed. D. Elmar L. Cabrera

LEARNING COMPETENCY:

• Illustrate the following measures of position: quartiles,

deciles, and percentiles. (M10SP - IVa - 1 )

Pre-assessment

Directions: Read each item carefully. Write the letter that corresponds to the correct answer in your activity notebook/answer sheet.

1. The median score is also the _____________. A. 75th percentile B. 5th decile C. 3rd decile D. 1st quartile
2. When a distribution is divided into hundred equal parts, each score point that describes the distribution is called a ___________. A. percentile B. decile C. quartile D. median
3. The lower quartile is equal to ______________. A. 50th percentile B. 25th percentile C. 2nd decile D. 3rd^ quartile
4. Rochelle got a score of 55 which is equivalent to 70th percentile in a mathematics test. Which of the following is NOT true? A. She scored above 70% of her classmates. B. Thirty percent of the class got scores of 55 and above. C. If the passing mark is the first quartile, she passed the test. D. Her score is below the 5th decile.
5. In a 100-item test, the passing mark is the 3rd quartile. What does it imply? A. The students should answer at least 75 items correctly to pass the test. B. The students should answer at least 50 items correctly to pass the test. C. The students should answer at most 75 items correctly to pass the test. D. The students should answer at most 50 items correctly to pass the I I

OBJECTIVES:

K: Define the measures of position: quartiles, deciles, and

percentiles.

S : Illustrate quartiles, deciles, and percentiles.

: Calculate specified measure of position for ungrouped

data.

A: Perform the given task with accuracy.

Did you take the National Career Assessment Examination (NCAE) when you were in Grade 9? If so, what was your score? Did you know your rank? Have you thought of comparing your academic performance with that of your classmates? Have you wondered what score you need for each subject area to qualify for honors? Have you asked yourself why a certain examinee in any national examination gets higher rank than the other examinees? Some state colleges and universities are offering scholarship programs for graduating students who belong to the upper 5%, 10%, or even 25%. What does this mean to you? In this module, you will study about the following measures of position: quartiles, deciles, and percentiles. Be ready to answer the following questions:

1. How would I know my position given the academic rank?
2. What are the ways to determine the measure of position in a set of data? Lesson Illustrating the Following Measures of Position: Quartiles, Deciles and Percentiles and percentiles. Ezra Louis- March 1, 2018

Discussion:

➢ How do you define Measures of Position? Answer: They are techniques that divide a set of data into equal groups. ➢ What are the different measures of position? Answers: Quartiles, Deciles, Percentiles ➢ Can you define Quartiles? Quartiles are points that divide the ranked data into four equal parts. Each set of data has three quartiles. 25% 25% 25% 25% L Q 1 Q 2 Q 3 H L = lowest score Q 1 = First quartile or lower quartile Q 2 = second quartile or middle quartile Q 3 = third quartile or upper quartile H = highest score

1. First quartile (Q 1 ) is the value in the data set such that 25% of the data points are less than this value and 75% of the data set is greater than this value.
2. Second quartile ( Q 2 ) is the value in the data set such that 50% of the data points are less than this value and 50% of the data set are greater than this value.
3. Third quartile (Q 3 ) is the value such that 75% of the values are less than this value and 25% of the values are greater than this value.
4. Interquartile range is the difference between the upper quartile (Q 3 ) and the lower quartile (Q 1 ) in a set of data. is It

The Quartile for Ungrouped Data Illustrative examples Example 1. A group of students obtained the following scores in their statistics quiz: 8 , 2 , 5 , 4 , 8 , 5 , 7 , 1 , 3 , 6 , 9 Solution: Observe how the lower quartile (Q 1 ), middle quartile (Q 2 ), and upper quartile (Q 3 ) of the scores are obtained. Complete the statements below: ➢ The first quartile 3 is obtained by observing the position of 3 which is in the middle of the numbers from 1 to 5. (observe the position of 3 from 1 to 5) ➢ The second quartile, 5 , is obtained by observing the position of 5 which is in the middle of the numbers from 1 to 9. (observe the position of 5 from 1 to 9) ➢ The third quartile, 8 , is obtained by observing the position of 8 which is in the middle of the numbers from 6 to 9. (observe the position of 8 from 6 to 9). Mendenhall and Sincich Method Using Statistics for Engineering and the Sciences, define a different method of finding quartile values. To apply their method on a data set with n elements, first calculate: Lower Quartile (L) = Position of Q 1 = 1 4 (n + 1) and round to the nearest integer. If L falls halfway between two integers, round up. The Lth element is the lower quartile value (Q 1 ). Next calculate: Upper Quartile (U) = Position of Q 3 = 3 4 (n + 1) and round to the nearest integer. If U falls halfway between two integers, round down. The Uth element is the upper quartile value (Q 3 ). Example 2. Example data set: {1, 3, 7, 7, 16, 21, 27, 30, 31} and n = 9. To find Q 1 , locate its position using the formula Q 1 = 1 4 (n + 1) and round off to the nearest integer. Position of Q 1 = 1 4 (n + 1) = 1 4

= 1 4

The computed value 2.5 becomes 3 after rounding up. The lower quartile value (Q 1 ) is the 3rd data element, so Q 1 = 7.

Step 3: Add the result in step 2, (1.5), to the 7th or smaller number. 27 + 1.5 = 28. Therefore, the value of Q3 = 28. The Deciles for Ungrouped Data The deciles are the nine score points which divide a distribution into ten equal parts. They are deciles and are denoted as D 1 , D 2 , D 3 ,…, D 9. They are computed in the same way that the quartiles are calculated. Illustrative example Find the 3rd decile or D 3 of the following test scores of a random sample of ten students: 35 , 42 , 40 , 28 , 15 , 23 , 33 , 20 , 18 and 28. Solution: First, arrange the scores in ascending order. 15 18 20 23 28 28 33 35 40 42 n = 10 Second, find decile value on a data with n elements: To find its D 3 position, use the formula 3 10 (n + 1) and round off to the nearest integer Position of D 3 = 3 10

3 10

33 10

D 3 is the 3rd element. Therefore, D 3 = 20. The Percentile for Ungrouped Data The percentiles are the ninety-nine score points which divide a distribution into one hundred equal parts, so that each part represents the data set. They are used to characterize values according to the percentage below them. For example, the first percentile (P 1 ) separates the lowest 1% from the other 99%, the second percentile (P 2 ) separates the lowest 2% from the other 98%, and so on. The percentiles determine the value for 1%, 2%,…, and 99% of the data. P 30 or 30th^ percentile of the data means 30% of the data have values less than or equal to P 30. The 1 st^ decile is the 10th percentile (P 10 ). It means 10% of the data is less than or equal to the value of P 10 or D 1 , and so on.

Illustrative example: Find the 30th percentile or P 30 of the following test scores of a random sample of ten students: 35, 42, 40, 28, 15, 23, 33, 20, 18, and 28 n = 10 , k = 30 Solution: Arrange the scores from the lowest to the highest. 15 18 20 23 28 28 33 35 40 42 Steps to find percentile value on a data with n elements: To find its P 30 position use the formula 𝑘 (𝑛+ 1 ) 100 n and round off to the nearest integer. Position of P 30 = 30 ( 10 + 1 ) 100

30 ( 11 ) 100

330 100

P 30 is the 3rd element. Therefore, P 30 = 20 15 18 20 23 28 28 33 35 40 42 ACTIVITY: Solve the following problems:

1. Albert has an assignment to randomly ask 10 students in their school about their ages. The data are given in the table below. Name Age Name Age Ana 10 Tony 11 Ira 13 Lito 14 Susan 14 Christian 13 Antonette 13 Michael 15 Gladys 15 Dennis 12 a. What is Q 1 , Q 2 , and Q 3 of their ages? b. How many students belong to Q 1 , Q 2 , and Q 3 in terms of their ages?
2. Mrs. Lim gave a test to her students in Statistics. The students finished their test in 35 minutes. This time is the 2.5th decile of the allotted time. What does this mean?
3. Given 30 enumeration and multiple-choice items in their summative test in Mathematics. a. Find the 75th^ percentile or P 75 of the following test scores of 15 students from Grade 10 - A 22, 10, 13, 13, 20, 13, 11, 17, 18, 14, 21, 15, 21, 16, 15 b. How many students have passed the exam? ’s More

2. Mrs. Marana is a veterinarian. One morning, she asked her secretary to record the service time for 15 customers. The following are service times in minutes. 20, 35, 55, 28, 46, 32, 25, 56, 55, 28, 37, 60, 47, 52, 17 Find the value of the 2nd decile, 6th decile, and 8th^ decile.
3. The scores of Miss World candidates from seven judges were recorded as follows: 8.45, 9.20, 8.56, 9.13, 8.67, 8.85, and 9.17. a. Find the 60th^ percentile or P 60 of the judges’ scores. b. What is the P 35 of the judges’ scores? Complete the Cross Quantile Puzzle by finding the specified measures of position. Use linear interpolation. (In filling the boxes, disregard the decimal point. For example, 14.3 should be written as Given: Scores 5, 7, 12, 14, 15, 22, 25, 30, 36, 42, 53, 65

Additional Activities Across

2. D 8
3. 65 (𝑛+ 1 ) 100
4. 90 (𝑛+ 1 ) 100
5. P 9 Down
6. Q 2
7. 90 (𝑛+ 1 ) 100
8. P 40
9. P 52
10. P 54

WHAT I KNOW B 1. A 2. B 3. D 4. A 5. WHAT’S IN 7 1. 8 2. 32 3.

43 5. WHAT’S NEW Answers May Vary WHAT’S MORE , 13 =^2 𝑄 ; 75. 11 =^1 𝑄 a. 1.

25. 14 =^3 𝑄 ; 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 6 =^2 𝑄 ; 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 3 =^1 𝑄 b. 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 8 =^3 𝑄 learners This means that 25% of the 2. finished the test. A low quartile considered good, because it means the students finished the test in a short period of time. 20 =^75 𝑃 a. 3. 4 b. WHAT I HAVE LEARNED True 1. False 2. True 3. True 4. False 5. O WHAT I CAN D ASSESSMENT 14 =^3 𝑄 ; 9 =^1 𝑄 a. 1. is 13^1 𝑄 Below b. customers while is 31^3 𝑄 above customers hour th hour, 5 th 6 c. ; 25 =^2 𝐷 2. ; 47 =^6 𝐷 55 =^8 𝐷 a. 9.13 3. b. 8. ACTIVITY AL ADDITION