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10
Mathematics
Department of Education Republic of the Philippines
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Learner’s Module
Unit 4
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Mathematics – Grade 10 Learner’s Module
First Edition 2015
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Published by the Department of Education Secretary: Br. Armin A. Luistro FSC
Undersecretary: Dina S. Ocampo, PhD
Printed in the Philippines by REX Book Store
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Telefax: (02) 634-1054, 634-1072 E-mail Address: imcsetd@yahoo.com
Development Team of the Learner’s Module
Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and
Rosemarievic Villena-Diaz, PhD
Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz,
Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida
Editor: Maxima J. Acelajado, PhD
Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P.
Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD
Illustrator: Cyrell T. Navarro
Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan
Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr.,
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Introduction
This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students.
In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants.
There are eight (8) modules in this material. Module 1 – Sequences
Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions
Module 4 – Circles
Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position
With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills.
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Unit 4
Module 8: Measures of Position
... 355Lessons and Coverage ... 357
Module Map ... 357
Pre-Assessment ... 358
Learning Goals and Targets ... 361
Lesson 1: Measures of Position for Ungrouped Data ... 362
Activity 1 ... 362 Activity 2 ... 363 Activity 3 ... 363 Activity 4 ... 364 Activity 5 ... 369 Activity 6 ... 371 Activity 7 ... 371 Activity 8 ... 372 Activity 9 ... 372 Activity 10 ... 375 Activity 11 ... 375 Activity 12 ... 377 Activity 13 ... 378 Activity 14 ... 378 Activity 15 ... 379 Activity 16 ... 379 Activity 17 ... 380 Summary/Synthesis/Generalization ... 382
Lesson 2: Measures of Position for Grouped Data ... 383
Activity 1 ... 383 Activity 2 ... 384 Activity 3 ... 394 Activity 4 ... 395 Activity 5 ... 396 Activity 6 ... 396 Activity 7 ... 397 Activity 8 ... 398 Activity 9 ... 398 Activity 10 ... 401 Activity 11 ... 401 Summary/Synthesis/Generalization ... 402 Glossary of Terms ... 403
References and Website Links Used in this Module ... 403
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I. INTRODUCTION
Look at the pictures shown below. Do you recognize them? 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?
Whenever your teacher asks your class to form a line according to your height, what is your position in relation to your classmates?
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?
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In this module, you will study about the measures of position. Remember to look for the answers to 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?
The basic purpose of all the measures of central tendency discussed so far during your Grade 7 and Grade 8 classes was to gain more knowledge and deeper understanding about the characteristics of a data set. Another method to analyze a data set is by arranging all the observations in either ascending or descending order of their magnitude. Then, this ordered set is divided into two equal parts by applying the concept of median. However, to have more knowledge about the data set, we may divide it into more parts of equal sizes. The measures of central tendency which are used for dividing the data into several equal parts are called partition values.
We shall discuss data analysis by dividing it into four, ten, and hundred parts of equal sizes and the corresponding partition values are called quartiles, deciles, and percentiles. All these values can be determined in the same way as the median. The only difference is in their location. Quantiles can be applied when:
1. dealing with large amount of data, which includes the timely results for standardized tests in schools, etc.
2. trying to discover the smallest as well as the largest values in a given distribution.
3. examining financial fields for academic as well as statistical studies. Quantiles are very useful because they help the government to find how the income in a country is distributed, how much of the total income is earned by low wage earning groups and by high wage earning groups. (If both groups earn the same proportion of the income, then there is income equality.)
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II. LESSONS AND COVERAGE
Lesson 1 – Measures of Position for Ungrouped Data Lesson 2 – Measures of Position for Grouped Data
In this lesson, you will learn to:
Lesson 1
Lesson 2
illustrate the following measures of position: quartiles, deciles, and percentiles.
calculate specified measure of position (e.g., 90th percentile) of a set of data.
interpret measures of position.
solve problems involving measures of position.
formulate statistical mini-research.
use appropriate measures of position and other statistical
methods in analyzing and interpreting research data.
Here is a simple map of the lessons in this entire module.
Study Tips
To do well in this particular topic, you need to remember and do the following: 1. Study each part of the module carefully.
2. Take note of all the formulas given in each lesson.
3. Have your own scientific calculator. Make sure you are familiar with the keys and functions of your calculator.
Solving Real-Life Problems Measures of Position Ungrouped Data Decile Percentile Quartile Grouped Data Decile Percentile Quartile
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III. PRE-ASSESSMENT
Part I.
Find out how much you already know about this module. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.
1. The median score is also the _____________. A. 75th percentile C. 3rd decile
B. 5th 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 C. quartile
B. decile D. median
3. The lower quartile is equal to ______________. A. 50th percentile C. 2nd decile B. 25th percentile 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 the set of scores: 14, 17, 10, 22, 19, 24, 8, 12, and 19, the median score is _______.
A. 17 C. 15
B. 16 D. 13
6. In a 70-item test, Melody got a score of 50 which is the third quartile. This means that:
A. she got the highest score.
B. her score is higher than 25% of her classmates. C. she surpassed 75% of her classmates.
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7. The 1st quartile of the ages of 250 fourth year students is 16 years old. Which of the following statements is true?
A. Most of the students are below 16 years old.
B. Seventy-five percent of the students are 16 years old and above. C. Twenty-five percent of the students are 16 years old.
D. One hundred fifty students are younger than 16 years.
8. 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 test.
9. In a group of 55 examinees taking the 50-item test, Rachel obtained a score of 38. This implies that her score is ______________.
A. below the 50th percentile C. the 55th percentile B. at the upper quartile D. below the 3rd decile 10. Consider the score distribution of 15 students given below:
83 72 87 79 82
77 80 73 86 81
79 82 79 74 74
The mean in the given score distribution of 15 students can also be interpreted as ______.
A. seven students scored higher than 79. B. seven students scored lower than 79.
C. seven students scored lower than 79 and seven students scored higher than 79.
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For items 11 to 14, refer to table A below.Table A
Score Frequency Cumulative Frequency Percentage (%) Cumulative
40-45 6 18 100.00
35-39 5 12 66.67
30-34 3 7 38.89
25-29 4 4 22.22
11. In solving for the 60th percentile, the lower boundary is ___.
A. 34 C. 39
B. 34.5 D. 39.5
12. What cumulative frequency should be used in solving for the 35th percentile? A. 4 C. 12 B. 7 D. 18 13. The 45th percentile is ________. A. 33.4 C. 30.8 B. 32.7 D. 35.6 14. The 50th percentile is _____. A. 36.0 C. 36.5 B. 37.0 D. 37.5 Part II.
Read and understand the situation below, then answer or perform what is asked. (6 points)
Jefferson, your classmate, who is also an SK Chairman in Barangay Cut-Cot, organized a Run for a Cause activity, titled FUN RUN. He informed your school principal to motivate students to join the said FUN RUN.
Conduct a mini-research or a simple research study on the students’ performance on the number of minutes it took them to reach the finish line.
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Teacher’s Rubric in Assessing Students’ Performance(Group Task) Standards 4 3 2 1 Understanding of Task Demonstrated substantial understanding of the content, processes, and demands of the task Demonstrated understanding of the content and task, even though some supporting ideas or details may have been overlooked or misunderstood Demonstrated gaps in their understanding of the content and task Demonstrated little understanding of the content Completion of Task Fully achieved the purpose of the task, including thoughtful, insightful interpretations, and conjectures Accomplished
the task Completed most of the task Attempted to accomplish the task, but with little or no success
Communication of findings
Communicated their ideas and findings effectively, raised interesting and provocative questions, and went beyond what was expected Communicated their findings effectively Communicated their ideas and findings
Did not finish the research study and/or were not able to communicate ideas very well
Group Process
Used all their time productively Everyone was involved and contributed to the group process and product. Worked well together most of the time They usually listened to each other and used each other’s ideas. Worked together some of the time Not everyone contributed equal efforts to the task.
Did not work very
productively as a group Not everyone contributed to the group effort.
IV. LEARNING GOALS AND TARGETS
After going through this module, you should be able to demonstrate understanding of key concepts of measures of position. Moreover, you should be able to conduct systematically a mini-research by applying the different statistical methods.
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Let us start our study of this module by first reviewing the concept of median, which is one of the concepts needed in the study of this module. Discuss the answers to the questions below with a partner.
The midpoint between two numbers x and y on the real number line is 2 x y . A B C x x y2 y
1. Find the coordinates of the midpoint (Q1) of AB in terms of x and y. A Q1 B x x y2
2. Find the coordinates of the midpoint (Q2) of BC in terms of x and y.
B Q2 C
x y2 y
3. In the given example, AC represents a distribution. What does point B represent in the distribution?
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The median divides the distribution into two equal parts. It is a point in the distribution where one-half of the distribution lies below it and one-half above it. One-half of the distribution lies below B and one-half lies above it. Hence, B represents the median.
Below is the RG2 worksheet which will determine your prior knowledge about the topic.
Answer the main question: What are the ways to determine the position in a
set of data? Write your answer in the Ready part of the RG2 Worksheet. RG2 Worksheet
Ready : Get set : Go :
Write your initial definition of the different measures of position. My Definition Table
Measures of Positions My Initial Definition
Quartile Decile
Percentile Activity 3:
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This part of the module enables you to understand quantiles in a set of ungrouped data. The activities in this section will help you answer the question, What are the ways to determine the measure of position in agiven set of data?
The understanding that you will gain in doing these activities will help you understand measures of position.
1. You are the fourth tallest student in a group of 10.
If you are the 4th tallest student, therefore 6 students are shorter than you.
It also means that 60% of the students are shorter than you. If you are the 8th tallest student in a group of 10, how many percent of the students are shorter than you?_________________________________ 2. A group of students obtained the following scores in their statistics
quiz:
8 , 2 , 5 , 4 , 8 , 5 , 7 , 1 , 3 , 6 , 9 Activity 4:
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Middle Quartile isalso the_______
First, arrange the scores in ascending order:
1 , 2 , 3 , 4 , 5 , 5 , 6 , 7 , 8 , 8 , 9
Observe how the lower quartile (Q1), middle quartile (Q2), and upper quartile (Q3) of the scores are obtained. Complete the statements below:
The first quartile 3 is obtained by ____________________________. (observe the position of 3 from 1 to 5)
The second quartile 5 is obtained by _________________________ . (observe the position of 5 from 1 to 9)
The third quartile 8 is obtained by ___________________________ . (observe the position of 8 from 6 to 9).
3. The scores of 10 students in a Mathematics seatwork are: 7 , 4 , 8 , 9 , 3 , 6 , 7 , 4 , 5 , 8
Arrange the scores in ascending order:
3 , 4 , 4 , 5 , 6 , 7 , 7 , 8 , 8 , 9
Discuss with your group mates:
a. your observations about the quartile. b. how each value was obtained.
c. your generalizations regarding your observations.
Q3 Upper quartile Q1 Lower quartile Q2 Middle quartile Q3 Upper quartile Q1 Lower quartile Q2 6 7 6.5 2
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Let us take a closer look at the quartiles.The Quartile for Ungrouped Data
The quartiles are the score points which divide a distribution into four equal parts. Twenty-five percent (25%) of the distribution are below the first quartile, fifty percent (50%) are below the second quartile, and seventy-five percent (75%) are below the third quartile. Q is called the lower quartile and 1
3
Q is the upper quartile. Q < 1 Q <2 Q , where 3 Q is nothing but the median. 2 The difference between Q and3 Q is the interquartile range. 1
Since the second quartile is equal to the median, the steps in the computation of median by identifying the median class is the same as the steps in identifying the Q1 class and the Q3 class.
a. 25% of the data has a value ≤ Q1 b. 50% of the data has a value ≤ X or Q2 c. 75% of the data has a value ≤ Q3
Example 1.
The owner of a coffee shop recorded the number of customers who came into his café each hour in a day. The results were 14, 10, 12, 9, 17, 5, 8, 9, 14, 10, and 11. Find the lower quartile and upper quartile of the data.
Solution:
In ascending order, the data are
5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17
The least value in the data is 5 and the greatest value in the data is 17.
The middle value in the data is 10.
The lower quartile is the value that is between the middle value and the least value in the data set.
So, the lower quartile is 9.
The upper quartile is the value that is between the middle value
and the greatest value in the data set.
So, the upper quartile is 14.
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Example 2.
Find the average of the lower quartile and the upper quartile of the data. Component Quantity hard disk 290 monitors 370 keyboards 260 mouse 180 speakers 430 Solution:
In increasing order, the data are 180, 260, 290, 370, 430.
The least value of the data is 180 and the greatest value of the data is 430.
The middle value of the data is 290.
The lower quartile is the value that is between the least value
and the middle value.
So, the lower quartile is 260.
The upper quartile is the value that is between the greatest value and the middle value.
So, the upper quartile is 370.
The average of the lower quartile and the higher quartile
is equal to 315.
Example 3.
The lower quartile of a data set is the 8th data value. How many data values are there in the data set?
Solution:
The lower quartile is the median data value of the lower half of the data set.
So, there are 7 data values before and after the lower quartile. So, the number of data values in the lower half is equal to
7+7+1.
The number of values in the data set is equal to lower half + upper half + 1.
The number of values in the lower and upper halves are equal. Formula:
15+15+1=31
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Another solution: 1 4 (n + 1) = 8 n + 1 = 32 n = 31 Example 4.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 1
1 1
4
Q n
and round to the nearest integer. If L falls halfway between two integers, round up. The Lth element is the lower quartile value (Q1).
Next calculate:
Upper Quartile (U) = Position of 3 3
1
4Q n
and round to the nearest integer. If U falls halfway between two integers, round down. The Uth element is the upper quartile value (Q3).
So for our example data set:
{1, 3 , 7, 7, 16 , 21, 27, 30 , 31} and n = 9.
To find Q1, locate its position using the formula 14
n 1
and round off to the nearest integer.Position of 1 1
1
4 Q n 41 (9 + 1) 1 4 (10) 2.5DEPED COPY
The computed value 2.5 becomes 3 after rounding up. The lower quartile value (Q1) is the 3rd data element, so Q1 = 7. Similarly:
Position of 3 3
1
4 Q n 3 9 1
4 3 10
4 = 7.5The computed value 7.5 becomes 7 after rounding down. The upper quartile value (Q3) is the 7th data element, so Q3 = 27.
Using this method, the upper quartile (Q3) and lower quartile (Q1)
values are always two of the data elements.
Find the first quartile (Q1), second quartile (Q2), and the third quartile (Q3), given the scores of 10 students in their Mathematics activity using Mendenhall and Sincich Method.
4 9 7 14 10 8 12 15 6 11
Example 5.
Find the first quartile (Q1) and the third quartile (Q3), given the scores of 10 students in their Mathematics activity using Linear Interpolation.
1 27 16 7 31 7 30 3 21
Solution:
a. First, arrange the scores in ascending order.
1 3 7 7 16 21 27 30 31
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b. Second, locate the position of the score in the distribution.Position of 1
1 1 4 Q n 41 9 1
41 10
= 2.5Since the result is a decimal number, interpolation is needed. c. Third, interpolate the value to obtain the 1st quartile.
Steps of Interpolation
Step 1: Subtract the 2nd data from the 3rd data. 7 – 3 = 4
Step 2: Multiply the result by the decimal part obtained in the second step (Position of Q1).
4(0.5) = 2
Step 3: Add the result in step 2, to the 2nd or smaller number.
3 + 2 = 5
Therefore, the value of Q1 = 5.
Solution:
a. First, arrange the scores in ascending order.
1 3 7 7 16 21 27 30 31
b. Second, locate the position of the score in the distribution. Position of 3
3 1 4 Q n 3 9 14
3104 = 7.5DEPED COPY
c. Third, interpolate the value to obtain the 3rd quartile.Steps of Interpolation
Step 1: Subtract the 7th data from the 8th data. 30 - 27 = 3
Step 2: Multiply the result by the decimal part obtained in the third step (Position of Q3).
3(0.5) = 1.5
Step 3: Add the result in step 2, (1.5), to the 7th or smaller number.
27 + 1.5 = 28.5 Therefore, the value of Q3 = 28.5
Note: As we can see, these methods sometimes (but not always) produce the same results.
Find the first quartile (Q1), second quartile (Q2), and the third quartile (Q3), given the scores of 10 students in their Mathematics activity using Linear Interpolation.
4 9 7 14 10
8 12 15 6 11
Albert has an assignment to ask at random 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 Activity 7: Activity 6:
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1. What is Q1, Q2, and Q3 of their ages?2. How many students belong to Q1, Q2,and Q3 in terms of their ages? 3. Have you realized the process of finding quartiles while doing the
activity?
Aqua Running has been promoted as a method for cardiovascular conditioning for the injured athlete as well as for others who desire a low impact aerobic workout. A study reported in the Journal of Sports Medicine investigated the relationship between exercise cadence and heart rate by measuring the heart rates of 20 healthy volunteers at a cadence of 48 cycles per minute (a cycle consisted of two steps).
The data are listed here:
87 109 79 80 96 95 90 92 96 98
101 91 78 112 94 98 94 107 81 96
Find the lower and upper quartiles of the data.
Consider the following nicotine levels of 40 smokers:
0 87 173 253 1 103 173 265 1 112 198 266 3 121 208 277 17 123 210 284 32 130 222 289 35 131 227 290 44 149 234 313 48 164 245 477 86 167 250 491 Find the lower and upper quartiles of the data.
Activity 9: Activity 8:
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The Deciles for Ungrouped DataThe deciles are the nine score points which divide a distribution into ten equal parts. They are deciles and are denoted as D1,D2, D3,…, D9. They are computed in the same way that the quartiles are calculated.
Example 6.
Find the 3rd decile or D3 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 Steps to find decile value on a data with n elements:
To find its D3 position, use the formula 103
n1
and round off to the nearest integer. Position of D3 103 10 1
103 11
3310 = 3.3 ≈ 3 D3 is the 3rd element. Therefore, D3 = 20. D1 D2 D3 D4 D5 D6 D7 D8 D9DEPED COPY
Example 7
1. Mrs. Labonete 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?
Explanation:
This means that 25% of the learners finished the test. A low quartile considered good, because it means the students finished the test in a short period of time.
2. Anthony is a secretary in one big company in Metro Manila. His salary is in the 7th decile. Should Anthony be glad about his salary or not? Explain your answer.
Solution:
70% of the employees receive a salary that is less than or equal to his salary and 30% of the employees receive a salary that is greater than his salary. Anthony should be pleased with his salary.
D2.5
35 minutes
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Mrs. Marasigan 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.
After studying several discussions, examples, and activities, it will be good for you to look back and check if there are still aspects which you find confusing and hard. You are now ready to answer questions like: How can the position of a certain value in a given set of data be described and used in solving real-life problems?
Given 50 multiple-choice items in their final test in Mathematics, the scores of 30 students are the following:
23 38 28 46 22 20 18 34 36 35
45 48 16 22 27 25 29 31 30 25
44 21 18 43 21 26 37 29 13 37
Calculate the following using the given data.
1. Q1 4. D2
2. Q2 5. D3
3. Q3 Activity 11:
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The Percentile for Ungrouped DataThe percentiles are the ninety-nine score points which divide a distribution into one hundred equal parts, so that each part represents the data set. It is used to characterize values according to the percentage below them. For example, the first percentile (P1) separates the lowest 1% from the other 99%, the second percentile (P2) separates the lowest 2% from the other 98%, and so on.
The percentiles determine the value for 1%, 2%,…, and 99% of the data. P30 or 30th percentile of the data means 30% of the data have values less than or equal to P30.
The 1st decile is the 10th percentile (P10). It means 10% of the data is less than or equal to the value of P10 or D1, and so on.
Example 8
Find the 30th percentile or P30 of the following test scores of a random sample of ten students: 35, 42, 40, 28, 15, 23, 33, 20, 18, and 28.
Solution:
Arrange the scores from the lowest to the highest. 15 18 20 23 28 28 33 35 40 42
Q1 Q2 Q3 P25 P50 P75
P10 P20 P30 P40 P50 P60 P70 P80 P90
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Steps to find percentile value on a data with n elements:To find its P30 position use the formula
1
100
k n
and round off to the nearest integer. Position of 30 30 10 1
100 P 30 11
100 300100 = 3.3 = 3.3 ≈ 3 P30 is the 3rd element. Therefore, P30 = 20.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.
1. Find the 60th percentile or P60 of the judges’ scores. 2. What is the P35 of the judges’ scores?
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Given a test in Calculus, the 75th percentile score is 15.What does it mean? What is its measure of position in relation to the other data?
Interpret the result and justify.
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 1 2 3 4 5 6 7 8 9 3 4 1 Activity 14: Activity 13: Across 2. D7 4. 65
1
100 n 8. 90
1
100 n 9. P9 Down 1. Q2 3. 90
1
100 n 5. P40 6. P52 7. P54DEPED COPY
This section of the module will test your understanding of the different measures of position by applying it to real-life situations. To demonstrate and apply your knowledge, you will be given a practical task specifically in the field of business and social sciences.
Write each step in finding the position / location in the given set of data using the cloud below. Add or delete clouds, if necessary.
A total of 8000 people visited a shopping mall over 12 hours.
Estimate the third quartile (when 75% of the visitors had arrived). Estimate the 40th percentile (when 40% of the visitors had arrived).
Time (hours) People 2 450 4 1500 6 2300 8 5700 10 6850 12 8000 Activity 16: Activity 15:
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Create a scenario of the task in paragraph form incorporating GRASPS.Goal: Make your own criteria in choosing the Cleanest Classroom
Role: Students by Section
Audience: The School Administration and Supreme Student Government Officers
Situation: The SSG Officers will reward a certificate of recognition to those who will rank 1st based on the given standards.
Product /Performance: Criteria
Standards: Understanding of task, completion of task, communication of findings, group process
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Teacher’s Rubric for Assessing Students’ Performance(Group Task)
Standards 4 3 2 1
Understanding
of Task Demonstrated a substantial understanding of the content, processes, and demands of the task Demonstrated understanding of the content and task, even though some supporting ideas or details may have been overlooked or misunderstood Demonstrated gaps in their understanding of the content and task Demonstrated minimal understanding of the content Completion of
Task Fully achieved the purpose of the task, including thoughtful, insightful, interpretations and conjectures Accomplished
the task Completed most of the task
Attempted to accomplish the task, but with little or no success
Communication
of findings Communicated their ideas and findings effectively, raised interesting and provocative questions, and went beyond what was expected Communicated their findings effectively Communicated their ideas and findings
Did not finish the
investigation and/or were not able to communicate ideas very well
Group Process Used all their
time productively Everyone was involved and contributed to the group process and product. Worked well together most of the time They usually listened to each other and used each other’s ideas. Worked together some of the time Not everyone contributed equal efforts to the task.
Did not work very productively as a group Not everyone contributed to the group effort.
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SUMMARY/SYNTHESIS/GENERALIZATION
This lesson was about measures of position for ungrouped data. The lesson provided you with opportunities to illustrate and compute for quartiles, deciles, and percentiles of ungrouped data. You were also given the opportunity to formulate and solve real-life problems involving measures of position.
You have learned the following:
Quartile for Ungrouped Data Position of
1
4 k k Q n Decile for Ungrouped Data Position of
1
10
k
k D n
Percentile for Ungrouped Data
Position of
1
100
k
k P n
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To check your readiness for the next topic, review the previous lessons. These will help you in the study of measure of position for grouped data. As you study the module, you may answer the question: How are measures of position for grouped data used in real-life situations? Do and accomplish the activities with your partner.
The following are scores of ten students in their 40-item quiz.
34 23 15 27 36 21 20 13 33 25
1. What are the scores of the students which are less than or equal to 25% of the data?
______________________________________________________ 2. What are the scores of the students which are less than or equal to
65% of the data?
______________________________________________________ 3. What are the scores of the students which are less than or equal to 8%
of the data?
______________________________________________________ Activity 1:
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Use your scientific calculator to answer the following questions. Do this activity as fast as you can.1. The bank manager observes the bank deposits in one specific day are as follows:
1150 5000 6500 1000 8500
9000 1200 1750 1100 4500
750 1500 1600 11 000 12 500 7000 9500 1200 13 500 1400 Find the 75th percentile.
2. The weights of the students in a class are the following: 69, 70, 75, 66, 83, 88, 66, 63, 61, 68, 73, 57, 52, 58, and 77.
Compute the 15th percentile.
3. Mr. Mel Santiago is the sales manager of JERRY’S Bookstore. He has 40 sales staff members who visit college professors all over the Philippines. Each Saturday morning, he requires his sales staff to send him a report. This report includes, among other things, the number of professors visited during the previous week. Listed below, ordered from smallest to largest, are the number of professors visited last week.
38 40 41 45 48 48 50 50 51 51 52 52 53 54 55 55 55 56 56 57 59 59 59 62 62 62 63 64 65 66 66 67 67 69 69 71 77 78 79 79
Determine the following. a. 3rd quartile
b. 9th decile c. 33rd percentile Activity 2:
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fQ1Did you find the previous activities easy? Were you able to answer it? Are you now ready to get the measures of position in a grouped data? To help you understand the next topic, notes with illustrative examples are provided.
The Quartile for Grouped Data
Recall that quartiles divide the distribution into four equal parts.
The steps in computing the median are similar to that of Q1 and Q3. In finding the median, we first need to determine the median class. In the same manner, the Q1 and the Q3 class must be determined first before computing for the value of Q1 and Q3. The Q1 class is the class interval where the
4 th
N score is contained, while the class interval that contains the
3 th
4
N
score is the Q3 class.
In computing the quartiles of grouped data, the following formula is used: 4 b k Qk kN cf Q LB i f
where: LB = lower boundary of the Qk class
N = total frequency
= cumulative frequency of the class before the Qk class
= frequency of the Qk class i = size of class interval
k = nth quartile, where n = 1, 2, and 3 b
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Example 1.
Calculate the Q1,Q2, and Q3 of the Mathematics test scores of 50 students.
Scores Frequency 46-50 4 41-45 8 36-40 11 31-35 9 26-30 12 21-25 6 Solution: Class Interval Scores Frequency (f) Boundaries Lower (LB) Less than Cumulative Frequency (<cf) 46-50 4 45.5 50 41-45 8 40.5 46 36-40 11 35.5 38 31-35 9 30.5 27 26-30 12 25.5 18 21-25 6 20.5 6 N = 50
i
Therefore, 25% of the students have a score less than or equal to 28.21.
(28th-38th score) Q3 class (19th-27th score) Q2 class (7th-18th score) Q1 class 1 1 4 b Q N cf Q LB i f 1 25.5 12.5 612 5 Q 1 28.21 Q 5 25.5 LB 50 N 6 b cf 2 12 Q f 50 4 4 12.5
N
This means we need to find the class interval where the 12.5th score is contained.
Note that the 7th-18th scores belong to the class interval: 26-30. So, the 12.5th score is also within the class interval.
The Q1 class is class interval 26-30.
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Therefore, 50% of the students have a score less than or equal to 34.39
Therefore, 75% of the students have a score less than or equal to 40.27. The third quartile 40.27 falls within the class boundaries of 36-40 which is (35.5-40.5) Q2 class: 24N
2 50 4This means we need to find the class interval where the 25th score is contained.
Note that the 19th-27th scores belong to the class interval: 31-35. So, the 25th score is also within the class interval.
The Q2 class is the class interval 31-35. 3 3 3 4 b Q N cf Q LB i f 3 37.5 27 35.5 5 11 Q 3 40.27 Q Q3 class: 34N
This means we need to find the class interval where the 37.5th score is contained.
Note that the 28th-38th scores belong to the class interval: 36-40. So, the 37.5th score is also within the class interval.
The Q3 class is class interval 36-40. 2 2 2 4 b Q N cf Q LB i f 2 25 18 30.5 5 9 Q 2 34.39 Q 30.5 LB 50 N 18 b cf 2 9 Q f 5 i 35.5 LB 50 N 27 b cf 2 11 Q f 5 i 100 4 25
3 50 4 150 4 37.5DEPED COPY
The Deciles for Grouped DataDeciles are those values that divide the total frequency into 10 equal parts. The kth decile denoted by D is computed as follows: k
10 k b k D kN cf D LB i f
where: LB = lower boundary of the Dk class
N = total frequency
cfb = cumulative frequency before the Dk class
k D
f = frequency of the Dk class
i = size of class interval
k = nth decile where n = 1, 2, 3, 4, 5, 6, 7, 8, and 9
Example 2.
Calculate the 7th decile of the Mathematics test scores of 50 students. Scores Frequency 46-50 4 41-45 8 36-40 11 31-35 9 26-30 12 21-25 6
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Solution: Class Interval Scores Frequency (f) Lower Boundaries (LB) Less than Cumulative Frequency (<cf) 46-50 4 45.5 50 41-45 8 40.5 46 36-40 11 35.5 38 31-35 9 30.5 27 26-30 12 25.5 18 21-25 6 20.5 6 N = 50Therefore, the 7th decile is equivalent to the 70th percentile. Therefore, 70% of the students got a score less than or equal to 39.14.
(28th-38th score) D7 class 7 7 7 10 b D N cf D LB i f 7 35.5 35 2711 5 D 7 39.14
D
D7 class: 7 10 N = = 35010 = 35This means we need to find the class interval where the 35th score is contained.
Note that the 28th-38th scores belong to the class interval: 36-40. So, the 35th score is also within the class interval.
The D7 class is the class interval 36-40.
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The Percentile for Grouped DataThe percentile of grouped data is used to characterize values according to the percentage below them.
Early on, you have already learned that kth quartile denoted by Qk and
the kth deciles denoted by Dk are computed, respectively, as follows:
4 k b k Q kN cf Q LB i f and 10 k b k D kN cf D LB i f
Finding percentiles of a grouped data is similar to that of finding quartiles and deciles of a grouped data.
The kth percentile, denoted by Pk, is computed as follows:
100 k b k P kN cf P LB i f where:
LB = lower boundary of the kth percentile class N = total frequency
cfb = cumulative frequency before the percentile class k
P
f = frequency of the percentile class
i = size of class interval
k = nth percentile where n = 1, 2, 3,…, 97, 98, and 99
Example 3.
Calculate the 65th percentile and 32nd percentile of the Mathematics test scores of 50 students.
Scores Frequency 46-50 4 41-45 8 36-40 11 31-35 9 26-30 12 21-25 6
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(28th-38th score) P65 class (7th-18th score) Q1 class Solution: i = 5Therefore, 65% of the students got a score less than or equal to 36-40. Class Interval Scores Frequency (f) Lower Boundaries (LB) Less than Cumulative Frequency (<cf) 46-50 4 45.5 50 41-45 8 40.5 46 36-40 11 35.5 38 31-35 9 30.5 27 26-30 12 25.5 18 21-25 6 20.5 6 65 65 65 100 b P N cf P LB i f 65 32.5 27 35.5 5 11 P 65 38 P P65 class : 65100N = = 3250100 = 32.5
This means we need to find the class interval where the 32.5th score is contained.
Note that the 28th-38th scores belong to the class interval: 36-40. So, the 32.5th score is also within the class interval.
The P65 class is the class interval 36-40. 65 35.5 50 27 11 b P LB N Cf f
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Therefore, 32% of the students got a score less than or equal to 29-67. Percentile RankPercentile ranks are particularly useful in relating individual scores to their positions in the entire group. A percentile rank is typically defined as the proportion of scores in a distribution that a specific score is greater than or equal to. For instance, if you received a score of 95 on a mathematics test and this score was greater than or equal to the scores of 88% of the students taking the test, then your percentile rank would be 88.
An example is the National Career Assessment Examination (NCAE) given to Grade 9 students. The scores of students are represented by their percentile ranks.
100 P PR P P LB f P cf N iwhere: PR = percentile rank, the answer will be a percentage
P
cf = cumulative frequency of all the values below the critical value
P = raw score or value for which one wants to find a percentile rank
LB = lower boundary of the kth percentile class N = total frequency
i = size of the class interval
32 32 32 100 b P N cf P LB i f 32 16 6 25.5 5 12 P 32 29.67 P P32 class: 32100N =
32 50 100 = 1600100 = 16This means we need to find the class interval where the 16th score is contained.
Note that the 7th-18th scores belong to the class interval: 26-30. So, the 16th score is also within the class interval.
The P32 class is class interval 26-30.
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Example 4.
Find how many percent of the scores are greater than the cumulative frequency of 38 in the previous table.
Solution: Scores Frequency cf 46-50 4 50 41-45 8 46 36-40 11 38 31-35 9 27 26-30 12 18 21-25 6 6
100 P PR P P LB f P cf N iTherefore, 65% of the scores are less than the cumulative frequency of 38, while 35% of the scores are greater than the cumulative frequency of 38.
Example 5.
Assume that a researcher wanted to know the percentage of consultants who made Php5,400 or more per day.
Consultant Fees (in Php)
Number of
Consultants Cumulative Frequency
6400 – 7599 24 120 5200 – 6399 36 96 4000 – 5199 19 60 2800 – 3999 26 41 1600 – 2799 15 15 (28th– 38th score) 38 is within 36-40 LB = 35.5 P = 38 N = 50 fP = 11 P cf = 27 I = 5
38 35.5 27 100 27 50 5 PR P 65 PR P DEPED COPY
Round off the resulting value to the nearest whole number.Therefore, 55% of consultants make Php 5,400.00 or less per day and 45% of consultants make Php 5,400.00 or more per day.
Daily allowance of 60 students Class Interval f <cf 81-90 7 60 71-80 10 53 61-70 15 43 51-60 4 28 41-50 12 24 31-40 6 12 21-30 3 6 11-20 2 3 1-10 1 1 D6 P15 P35 D8 D4 P70 Q1 Q2 D8 Q3 ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ Activity 3: Php 5,400 is within 5200-6399 LB = 5199.5 N = 120 P = 5,400.00 cf P = 60 f P = 36 i = 1200
5400 5199.5 36 100 60 120 1200 PR
100 P PR P P LB f P cf N i 55.01 PR PDEPED COPY
Given the frequency distribution, compute for each quantile and match it with the letter code of its corresponding value to complete the phrase in the preceding page: Q1 T. 43 Q2 I. 61.83 Q3 N. 72.5 P15 Y. 35.5 P35 L. 48 P70 A. 69.83 D6 M. 65.83 D4 C. 50/5 D8 O. 75.5 R. 34
The following is a distribution for the number of employees in 45 companies belonging to a certain industry. Calculate the third quartile, 85th percentile, and 4th decile of the number of employees given the number of companies.
Number of
Employees Companies Number of
41 – 45 11 36 – 40 6 31 – 35 9 26 – 30 7 21 – 25 8 16 – 20 4 Activity 4:
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Find the 1st quartile, 7th decile, 35th percentile, and percentile rank of 115 and 155 for the following distribution.Class Interval Frequency
151 – 160 8 141 – 150 12 131 – 140 6 121 – 130 10 111 – 120 7 101 – 110 11 91 – 100 13 81 – 90 9 71 – 80 4
After having several discussions, examples, and activities, you need to have a closer look once again if there are still aspects which you find confusing and hard. You are now ready to answer questions like: How can the position of data be described and used in solving real-life problems?
Dennis and Christine scored 32 and 23, respectively, in the National Career Assessment Examination (NCAE). The determining factor for a college scholarship is that a student’s score should be in the top 10% of the scores of his/her graduating class. The students in the graduating class obtained the following scores in the NCAE.
NCAE Scores f LB <cf 39 – 41 6 36 – 38 7 33 – 35 9 30 – 32 13 27 – 29 22 Activity 6: Activity 5:
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NCAE Scores f LB <cf 24 – 26 10 21 – 23 9 18 – 20 7 15 – 17 8 12 – 14 4 9 – 11 2 6 – 8 1 3 – 5 11. Complete the table by filling in the values of LB (lower boundaries) and <cf (less than cumulative frequency). Explain how you arrived at your answers.
2. Find the 3rd quartile, 72nd percentile, and the 8th decile of the set of data.
3. What is the percentile rank of Dennis and Christine?
4. Based on their percentile and percentile ranks, will Dennis and Christine receive a scholarship? Explain your answer.
In this activity, you will be asked to complete the 1 – 4 – 3 chart. Write down what is being asked regarding the different measures of position.
1 – 4 – 3 LIST One thing I really love about this topic
1.
Four important reasons why I love this topic 1.
2. 3. 4.
Three things I still need to understand about this topic 1.
2. 3.
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You have already learned and identified the measures of position and the process of computing and interpreting results. You will now take a closer look at some aspects of the topic and check if you still have misconceptions about measures of position.
Write a reflection journal titled “Measure of Position” using the format: I. Things Learned and Insights
II. Concept Map III. Difficulties
IV. Unforgettable Experiences / Activities
Conduct a mini-research on students’ performance in the final examination in Mathematics. Apply the knowledge and skills you have learned in this lesson to evaluate and interpret test results and to make/formulate meaningful decisions based on the results to resolve the difficulties of the students.
Activity 9: Activity 8:
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Teacher’s Scoring Rubric in Assessing Students’ Performance(Group Task)
Criteria Proficient Approaching Proficient Developing Beginning
INTEGRATION OF KNOWLEDGE The paper demonstrated that the student fully understands and has applied concepts learned in the course. Concepts are integrated into the writer’s own insights. The writer provides concluding remarks that show analysis and synthesis of ideas. The paper demonstrated that the
student, for the most part, understands and has applied concepts learned in the course. Some of the conclusions, however, are not supported in the body of the paper. The paper demonstrated that the student, to a certain extent, understands and has applied concepts learned in the course.
The paper did not demonstrate that the student has fully understood and applied concepts learned in the course. TOPIC FOCUS The topic is focused narrowly enough for the scope of this assignment. The research study provides direction for the paper, either by statement of a position or hypothesis. The topic is focused but lacks direction. The paper is about a specific topic but the writer has not established a position.
The topic is too broad for the scope of this assignment. The topic is not clearly defined. DEPTH OF DISCUSSION In-depth discussion is evident in all In-depth discussion is evident in most The student has omitted pertinent The paper lacks in-depth discussion as
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Criteria Proficient Approaching Proficient Developing Beginningsections of the
paper sections of the paper content or content run-ons excessively. evidenced by cursory discussion (with limited supporting points) in all sections of the paper. COHESIVENESS Ties together information from all sources Paper flows from one issue to the next without the need for headings. Student's writing demonstrates an understanding of the relationship among materials obtained from all sources.
For the most part, ties together information from all sources Paper flows with only some disjointedness. Student's writing demonstrates an understanding of the relationship among materials obtained from all sources. Sometimes ties together information from all sources Paper did not flow - disjointedness is apparent. Student's writing does not demonstrate an understanding of the relationship among materials obtained from all sources.
Does not tie together information Paper does not have a good flow and appears to be created from disparate issues. Headings are necessary to link concepts. Writing does not demonstrate understanding of any relationships. SPELLING & GRAMMAR No spelling and/or grammar mistakes Minimal spelling and/or grammar mistakes Noticeable spelling and grammar mistakes Unacceptable number of spelling and/or grammar mistakes
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Ask your classmates about their Science, English, and Mathematics grades. Gather all the data from your classmates by listing. Then, construct a frequency distribution of a grouped data. (use i = 5).
Calculate the following: a. 1st quartile b. 2nd quartile c. 3rd quartile d. 7th decile e. 4th decile f. 60th percentile g. 85th percentile h. percentile rank of 75 i. percentile rank of 82 Interpret each result.
______________________________________________________________ ______________________________________________________________ ______________________________________________________________ _____________________________________________________________.
Write a good definition of the different measures of position. My Definition Table
Measures of Positions My Definition
Quartile
Decile Percentile
Activity 11: Activity 10:
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SUMMARY/SYNTHESIS/GENERALIZATION
In this lesson, you were able to illustrate measures of position for grouped data: quartiles, deciles, and percentiles, calculate a specified measure of position (e.g., 90th percentile) of a set of data, interpret measures of position, and solve problems involving measures of position. More importantly, you were given the chance to formulate and solve real-life problems, and demonstrate your understanding of the lesson by doing some practical tasks.
You have learned the following: Quartile for Grouped Data
4 b k Qk kN cf Q LB i f
Decile for Grouped Data 10 k b k D kN cf D LB i f
Percentile for Grouped Data 100 k b k P kN cf P LB i f Percentile Rank
100 P PR P P LB f P cf N iDEPED COPY
GLOSSARY OF TERMS
Deciles - the nine score points which divide a distribution into ten equal parts. These deciles are denoted as D1, D2, D3,… D9.
Percentiles - the ninety-nine score points which divide a distribution into one hundred equal parts so that each part represents 1001 of the data set. They are used to characterize values according to the percentage below them. Quantiles - measures of positions that divide a distribution into four, ten, and hundred equal parts. Such measures of positions are quartiles, deciles, and percentiles.
Quartiles - the score points which divide a distribution into four equal parts. Twenty-five percent (25%) of the distribution fall below the first quartile, fifty percent (50%) fall below the second quartile, and seventy-five percent (75%) fall below the third quartile.
REFERENCES AND WEBSITE LINKS USED IN THIS MODULE References:
De Guzman-Santos, R., De Guzman, T., Ungriano, A., Yabut, E. (2006).
Statistics. Manila, Philippines. Centro Escolar University Publishing
House.
Febre, Francisco Jr. (1987). Introduction to Staistics. Quezon City, Philippines. Phoenix Publishing House, Inc.
Manansala, T. (2007). Statistics. Manila, Philippines. Jimcy Publishing House. Mendenhall, W., Beaver, R., Beaver, B. (2006). Probability and Statistics.
Thomson Learning Asia.
Oronce, O., Mendoza, M. (2010). E-math IV. Quezon City, Philippines. Rex Book Store, Inc.
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Website Links as References and Sources of Learning Activities:http://www.slideshare.net/maggiev/the-interpretation-of-quartiles-and-percentiles-july-2009
This site provides formula, examples, and exercises of quartile, percentile, and decile.
http://www.mathsisfun.com/data/quartiles.html This site provides examples of quartile.
www.mathsisfun.com/data/percentiles.html
This site provides examples and exercises of percentile.
www.harding.edu/sbreezeel/460%20files/statbook/chapter5.pdf
This site provides formula, examples, and exercises of percentile and percentile ranks.
http://www.onlinemathlearning.com/quartile.html
This site provides problems for the cross-quantile problem. https://www.google.com.ph
The following sites provide pictures that made the module more attractive and interesting especially to students.
http://books.google.com.ph//
International Business Research By Neelankavil This provides exercise for business in calculator drill.
http://alstatr.blogspot.com/2013/06/quartiles-deciles-and-percentiles.html This provides exercise for business in calculator drill.
http://answers.yahoo.com/question/index?qid=20100630123126AA7lZZa This provides exercise for business in calculator drill.
http://www.icoachmath.com/problems/problemslink.aspx This site provides examples and exercises of quartile.
