Mathematics 10 Learner’s Material Unit 4

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10
Mathematics
Department of Education
Republic of the Philippines
This book was collaboratively developed and reviewed by
educators from public and private schools, colleges, and/or universities.
We encourage teachers and other education stakeholders to email their
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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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Development Team of the Learner’s Module
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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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electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

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Unit 4
Module 8: Measures of Position............................................................ 355
Lessons 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
Table of Contents
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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 PercentileQuartile
Grouped
Data
Decile PercentileQuartile
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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.
D. seventy-five percent of the class did not pass the test.
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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.
D. fourteen students scored lower than 79.
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For items 11 to 14, refer to table A below.
Table A
Score Frequency
Cumulative
Frequency
Cumulative
Percentage (%)
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
2
x y
y
1. Find the coordinates of the midpoint (Q1) of AB in terms of x and y.
A Q1 B
  
x
2
x y
2. Find the coordinates of the midpoint (Q2) of BC in terms of x and y.
B Q2 C
  
2
x y
y
3. In the given example, AC represents a distribution. What does point B
represent in the distribution?
Activity 1:
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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:
Activity 2:
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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 a
given 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 is
also 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. 1
Q is called the lower quartile and
3
Q is the upper quartile. 1
Q < 2
Q < 3
Q , where 2
Q is nothing but the median.
The difference between 3
Q and 1
Q is the interquartile range.
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.
Q1 Q2 Q3
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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
 So, the data set contains 31 data values.
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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
4
Q 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  1
1
4
n  and round off
to the nearest integer.
Position of   1
1
1
4
Q n
1
4
 (9 + 1)

1
4
(10)
 2.5
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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

= 7.5
The 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
Activity 5:
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b. Second, locate the position of the score in the distribution.
Position of  1
1
1
4
Q n 
 1
9 1
4
 
 1
10
4

= 2.5
Since 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 1
4
 
3
10
4

= 7.5
Since the result is a decimal number, interpolation is needed.
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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 Data
The 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  3
1
10
n  and round off to the
nearest integer.
Position of  3
3
10 1
10
D  
 3
11
10

33
10

= 3.3 ≈ 3
D3 is the 3rd element.
Therefore, D3 = 20.
D1 D2 D3 D4 D5 D6 D7 D8 D9
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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:
Activity 10:
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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. 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
D1 D2 D3 D4 D5 D6 D7 D8 D9
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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

300
100

= 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?
Activity 12:
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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
341
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. P54
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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
Activity 17:
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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
4k
k
Q n 
Decile for Ungrouped Data
Position of  1
10k
k
D n 
Percentile for Ungrouped Data
Position of  1
100k
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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1Q
f
Did 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
 
 
 
th
4
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
cf
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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)
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 = 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
12.5 6
25.5 5
12
Q
 
   
 
1
28.21Q 
 5
 25.5LB
 50N
 6b
cf

2
12Q
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.
Q1 class:
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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: 
2
4
N  2 50
4
This 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
40.27Q 
Q3 class:
3
4
N
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
34.39Q 
 30.5LB
 50N
 18b
cf

2
9Q
f
 5i
 35.5LB
 50N
 27b
cf

2
11Q
f
 5i


100
4
25
 


3 50
4
150
4
37.5
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The Deciles for Grouped Data
Deciles are those values that divide the total frequency into 10 equal
parts. The kth decile denoted by k
D is computed as follows:
10
k
b
k
D
kN
cf
D LB i
f
 
 
   
  
 
where: LB = lower boundary of the Dk class
N = total frequency
b
cf = cumulative frequency before the Dk class
kD
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 = 50
Therefore, 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 27
35.5 5
11
D
7
39.14D 
D7 class:
7
10
N
=
=
350
10
= 35
This 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 Data
The 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
b
cf = cumulative frequency before the percentile class
kP
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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41. D
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(28th-38th score)
P65 class
(7th-18th score)
Q1 class
Solution:
i = 5
Therefore, 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
38P 
P65 class :
65
100
N
=
=
3250
100
= 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 Rank
Percentile 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 i
where: 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
29.67P 
P32 class:
32
100
N
=
 32 50
100
=
1600
100
= 16
This 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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43. D
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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 i
Therefore, 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
P
f = 11
P
cf = 27
I = 5
  
   
  
38 35.5 27100
27
50 5PR
P
65PR
P 
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44. D
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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
P
cf = 60
P
f = 36
i = 1200
  
  
  
5400 5199.5 36100
60
120 1200PR
  
  
  
100 P
PR P
P LB f
P cf
N i
 55.01PR
P
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45. D
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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
Number of
Companies
41 – 45 11
36 – 40 6
31 – 35 9
26 – 30 7
21 – 25 8
16 – 20 4
Activity 4:
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46. D
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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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47. D
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397
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 1
1. 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.
Activity 7:
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48. D
EPED
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398
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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49. D
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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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50. D
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Criteria Proficient
Approaching
Proficient
Developing Beginning
sections 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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51. D
EPED
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401
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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52. D
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402
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 i
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53. D
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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
1
100
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.
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electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.