10
Mathematics
Quarter 4 – Module 2
Calculating
Measures of Positions
Department of Education. Republic of the Philippines
Mathematics – Grade 10
Alternative Delivery Mode
Quarter 4 – Module 2: Calculating Measures of Positions
First Edition, 2020
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Published by the Department of Education
Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Charlotte S. Tabaňag
Rhodel A. Lamban, PhD
Elbert R. Francisco, PhD
Alicia P. Micayabas, PhD
Manilen S. Lizano, PhD
Eleonor Villamor
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Randolph B. Tortola, PhD, CESO IV
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Assistant Schools Division Superintendent
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2
10
Mathematics
Quarter 4 - Module 2
Calculating
Measures of Positions
This instructional material 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
feedback, comments, and recommendations to the Department of Education
at bukidnon@deped.gov.ph.
We value your feedback and recommendations.
Department of Education. Republic of the Philippines
3
Table of Contents
COVER PAGE
Page
COPYRIGHT PAGE
TITLE PAGE
TABLE OF CONTENT
1
OVERVIEW
2-3
PRE-ASSESSMENT
4
Lesson 1: Measures of Position for Ungrouped Data
(Day 1 - 2)
Pre-Assessment
5
Presentation
6
Concept Development
6-11
Activities
11-12
Assessment
12
Application
13
Lesson 2: Measures of Positions for Grouped Data
(Day 3 - 5)
Pre-Assessment
14
Presentation
14
Concept Development
15-21
Activities
22
Assessment
22
GENERALIZATION/SYNTHESIS
23
APPLICATION
24
POST-ASSESSMENT
24-25
ANSWER KEYS
26
REFERENCES
27
1
OVERVIEW
This module was designed and written with you in mind. It is here to
help you master to calculate Measures of Positions. The scope of this
module permits it to be used in many different learning situations. The
language used recognizes your diverse vocabulary level. The lessons are
arranged for you to follow the standard sequence of the course. But the
order in which you read them can be changed to correspond with the
textbook you are now using.
The module is divided into two lessons, namely:


Lesson 1 – Measures of Position for Ungrouped Data
Lesson 2 – Measures of Position for Grouped Data
After going through this module, you are expected to:
1.
2.
3.
4.
5.
6.
Calculate
Calculate
Calculate
Calculate
Calculate
Calculate
quartiles in ungrouped data;
deciles in ungrouped data;
percentiles in ungrouped data;
quartiles in grouped data;
deciles in grouped data; and
percentiles in grouped data.
You need to study this module religiously. In a simple manner, it will
help you:
a. know your position in an academic rank;
b. deal with large amount of data, which includes test results for
Board exams;
c. determine the smallest and the largest values of the results of a
survey; and
d. analyze the statistical survey of the government or any large scales
values.
In this module, you will study on how to calculate on measures of
position (quartile, decile, and percentile) both on ungrouped and grouped
data. The purpose of studying on measures of central tendency and
variability in your previous grade was to earn more knowledge and
understanding on the characteristics of a set of data.
2
In measures of central tendency, a set of data was divided into two
equal parts to find its median. Then in this module we will divide the set of
data into several parts equally. We will divide it into four, ten, and hundred
parts. This equal parts are called partition values (quartiles, deciles, and
percentiles). These values can be determined same with finding the median
in the measures of central tendency and its only difference is their location.
Quartiles, deciles, and percentiles are called quantiles.
Quantiles can be applied when dealing with huge amount of data, which
includes the timely results for standardized tests in school, and by
discovering the smallest as well as the largest values in a given distribution,
and examining the financial fields for academe as well as statistical analysis.
It is also very useful in helping the government to find how the income in
the country distributed.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any
part of the module. Use a separate sheet of paper in answering
the exercises.
2. Don’t forget to answer Pre-Assessment before moving on to the
other activities included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking
your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are
through with it.
If you encounter any difficulty in answering the tasks in this module, do
not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone.
We hope that through this material, you will experience meaningful
learning and gain deep understanding of the relevant competencies. You
can do it!
3
PRE-ASSESSMENT
Choose the letter of the correct answer. Write the chosen letter on a separate sheet of
paper.
For items 1-3, refer to Charles Grade in 3rd Quarter: Charles have 91
in Eglish, 93 in Filipio, 90 in Mathematics, 90 in Science, 92 in Aral.Pan, 96 in
TLE, 94 in MAPEH, and 90 in EsP.
1. What is the average of Charles in 3rd quarter?
A. 90
B. 91
C. 92
D. 93
2. What grade of Charles during the 3rd quarter which frequently occur?
A. 90
B. 91
C. 93
D. 94
3. What must be the eight grade of Charles in order for him to get and
average grade of 94. Given 97, 92, 96, 91, 95, 94, 93, __.
A. 93
B. 94
C. 95
D. 96
For items 4 to 7, refer to Table below.
Score
Frequency
cf
LB
46-50
5
50
45.5
41-45
6
45
40.5
36-40
12
39
35.5
31-35
10
27
30.5
26-30
13
17
25.5
21-25
4
4
20.5
i= 5
∑N = 50
4. What cumulative frequency must be used in solving for the
A. 4
B. 10
C. 12
D. 13
5. The third quartile is _______________.
A. 36.875
B. 37.857
C.38.875
D. 39.875
6. Find the 20th percentile.
A. 26.81
B. 27.81
C 28.81
D. 29.81
7. The fifth decile is ______________.
A.31.5
B. 32.5
C.33.5
D. 34.5
4
?
Lesson
1
Calculating Measures of
Position for Ungrouped Data
To start this module, you should remember the concept of median in
the measures of central tendency.
This lesson was designed and written with you in mind. It is here to
help you master to calculate Measures of Positions. The language used
unrecognized your diverse vocabulary level. The lessons are arranged for you
to follow the standard sequence of the course.
After going through this module, you are expected to:
Calculate quantiles(quartile, decile, and percentile) in ungrouped data.
In finding the median for ungrouped data you will divide the data in two
equal parts. Then in this module, the data will be divided in several parts
equally by four (quartiles), ten(deciles), and a hundred(percentiles). Quartile,
decile, and percentile are called the Quantiles. Quantiles are in the
Measures of Position.
In a number line below, where can you find B?
X
A
y
B
C
,
To find the value of B, is by adding
where B is a midpoint
PRE-ASSESSMENT
In your midst
A group of learners having a score in their English quiz: 9, 6, 8, 8, 3, 4,
7, 11, and 10. In finding the median for ungrouped data you will divide the
data in two equal parts. What is the median?
5
PRESENTATION:
Ungrouped data
Ungrouped data is a set of data taken from a small group of people or
any group of an experiment. It is taken from an individual from a very small
group. Example is getting the average grade of the top 10 academic
performer in a science classroom. The data gathered was so small and it is
ungrouped data.
Then in this module, the data will be divided in several parts equally by
four (quartiles), ten(deciles), and a hundred(percentiles). Quartile, decile,
and percentile are called the Quantiles. Quantiles are in the Measures of
Position.
Measures of Positions for ungrouped data studied in a small group of
experiment. It is preferably in a small set of data. The best example is the
rank in an academic performance in a classroom, position of an elected
officials in a government, rank in an athletic through the number of minutes
performed, etc…
Quantiles are the quartiles, deciles, and percentiles. The data will be
divided into four for quartiles, into ten for deciles, and into hundred for
percentiles.
Table 1. Measures of Position
quantiles
Partition values
Quartiles
four
Deciles
ten
Percentiles
hundred
CONCEPT DEVELOPMENT
The Quartile for Ungrouped Data
The quartiles are the score points which divide a distribution into four
equal parts. Twenty-five (25%) per cent of the distribution are below the first
quartile, fifty (50%) per cent are below the second quartile, and seventy-five
(75%) per cent are below the third quartile.
is called the lower quartile
and
is called the upper quartile. <
< , where
is nothing but the
median. The difference between
and
is the interquartile range.
6
Let’s Explore:
For example: A one whole data which is divided into four parts is shown
in fig.1 and fig.2 below.
Fig. 1
Whole Data
This is a whole given data which need to be divided three times and has
four equal parts or measured in quartile. These are
, , and .
Fig. 2
a. 25% of the data has a value ≤
b. 50% of the data has a value ≤ x or
c. 75% of the data has a value ≤
The formula in finding quartiles are:
=
=
Where:
…..
=
=
k is the nth quartile,
n is the total number of data
is for first quartile,
is for second quartile; and
is for third quartile
For Example:
Find the lower and the upper quartile of the test scores of ten sample
learners: 15, 20, 25, 30, 28, 40, 44, 42, 34, and 30.
Solution: To find the unknown, follow the steps below.
7
Steps
Figures/Experiments
Step 1. Arrange the
scores/data in
ascending order.
You have… 15, 20, 25, 30, 15, 20, 25, 28, 30,
28, 40, 44, 42, 34, and 30. 30, 34, 40, 42, 44
Step 2. After arranging
the data What is the
smallest value?
15, 20, 25, 28, 30, 30, 34,
40, 42, 44
The least value is 15.
Step 3. After arranging
the data, What is the
greatest value?
15, 20, 25, 28, 30, 30, 34,
40, 42, 44
The greatest value is
44.
Step 4. Find
There are 10 values.
15, 20, 25, 28, 30, 30,
34, 40, 42, 44
position
using the formula
=
=
) ,,,,
=2.75 ≈ 3 or 3rd
position where it is 25
= 25
. Get the total
number of data and add
1 or (n+1). Then divide it
by 4.
Step 5. Find
position using the
formula
Answers
=
=2.75 ≈ 3 or 3rd position
where it is 25
Follow step 4.
=
=
) = 8.25
≈ 8 is a position and 8th
position is 40
15, 20, 25, 28, 30, 30,
34, 40, 42, 44
= ¾(10+1) = 8.25 ≈ 8
is a position and 8th
position is 40
= 40
If you are done proceed to the presentation on deciles. Congratulations!
The Decile for Ungrouped Data
The decile is the nine score points which divide a distribution into ten
equal parts. Denoted as
,
,
,
,
,
,
,
,
as presented in
fig.4. They are computed in the same way with the quartiles are calculated.
For example:
A one whole data which is divided into ten parts is shown in figure 1.
fig. 3
8
To find the decile, for ungrouped data, used this formula:
=
=
Where:
=
=
k is the nth quartile,
…
and so on…
n is the total number of data
is 1st decile,
is 2nd decile,
is 3rd decile,
is 4th decile,
is 5th decile,
is 6th decile,
is 7th decile,
is 8ht decile, and
is 9th decile.
For example:
Find the , and
of the test scores of ten sample learners: 15, 20, 25,
30, 28, 40, 44, 42, 34, and 30.
Solution:
Steps:
Solutions
Answers
Arrange the
scores in ascending
order
Your data: 15, 20, 25,
30, 28, 40, 44, 42, 34,
and 30.
15, 20, 25, 28, 30,
30, 34, 40, 42, 44
Find
position using
the formula
=
(10+1) =
(11) =
3.3 ≈ 3
= 3 or 3rd position
Find
position using
the formula
=
(10+1) =
(11) = 7.7≈ 8
= 8 or 8th position
15, 20, 25, 28, 30, 30,
34, 40, 42, 44
= 25
15, 20, 25, 28, 30, 30,
34, 40, 42, 44
= 40
How is your lesson? If you have difficulty on understanding this topic,
go back to the given example and solutions. If there is none, proceed to the
next topic on percentile.
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 according to percentage below their
values.
9
fig.4
This only shows the similarity of the calculations of the Quartiles,
Deciles, and Percentiles.
Fig.5
It shows that the first Decile ( ) is the 10th Percentile
. It separates
th
th
10 % of the rest of the data. The 5 decile ( ) or the 50 Percentile
separates the lowest 50% from the whole data and so on.
=
Formula:
=
=
………
=
Where:
k is the nth quartile,
is 10th percentile,
is 50th percentile,
n is the total number of data
is 30th percentile
is 99th percentile
For example:
Find the
10 students:
and
of the following test scores of a random sample of
42, 35, 41, 28, 33, 44, 28, 19, 21, and 25.
10
Solution: To find the answers, follow the steps below.
Steps
Solutions
Answers
Arrange the score from
lowest to highest
You have 42, 35, 41, 28, 33,
44, 28, 19, 21, and 25.
19,21,25,28,28,33,35
,41,42,44
Find the
using the
formula
=
=
=
= 4.4
= 4.4 ≈ 4, where it is the
element.
4th
Find the
using the
formula
19,21,25,28,28,33,35
,41,42,44
=
=
=
=
8.8
9th
= 8.8 ≈ 9, where it is the
element.
Therefore
= 28
19,21,25,28,28,33,35
,41,42,44
Therefore
= 42
How is your lesson? If you have no questions, answer Activity 1.1. Find Me!
ACTIVITIES:
Activity 1.1 Find Me!
Joeneth is a timekeeper of a mini-internet café. One day, she realized to
record the number of customers they will have for about seven days.
The following are the number of customers their shop
accommodated for seven days are 34, 28, 30, 33, 40, 44, and 40
Find the values of the
,
,
,
,
, and
Solutions are shown on the next page. Just follow the steps given.
11
had
Solutions:
To find the answers, follow the steps from the example.
Steps
Solutions
Answer
1.Arrange the given set of data.
After finding each quantiles it is not your final answer.
Remember we are tackling positions here. So, based your answers in
the arranged data.
2.To find
formula.
position, use the
=
=
=
3.To find
formula
position, use the
=
=
=
4.To find
formula
position, use the
=
=
=
5.To find
formula
position, use the
=
=
=
6.To find
formula
position, use the
=
=
=
7.To find
formula
position, use the
=
=
=
Do you have questions or queries? Let your teacher know. If you have
no questions, proceed to the Assessment.
ASSESSMENT
The grade 10 learner having their exercise during their MAPEH period
before weighing. Ten of them have their weights 75, 65. 70, 85, 72, 80, 67,
58, 77, and 65.
Calculate the following quantiles:
a.
b.
c.
d.
e.
f.
Congratulations! You are now ready to assess your understanding and
knowledge on calculating measures of position. You can answer Cross
Quantile Puzzle.
12
APPLICATION: Cross Quantile Puzzle
Complete the Puzzle by calculating the specified measures of Position.
In filling the boxes, disregard the decimal point. For example, the answer is
25.2 should be written in:
2
Given: Scores
and 45.4.
24.8,
2
55.3, 32.3,
Across:
Down:
5
2.)
1.)
35.8,
52.5,
4.)
3.)
46.3,
6.)
5.)
48.3,
7.)
8.)
1
8
2
7
3
5
4
6
How is the puzzle? Check your score.
If your score is below 6, goback to the given examples for you to
understand better the lesson.
If your score is six and above, proceed to lesson 2 in the next page.
13
Lesson
2
Measures of Position
(Grouped Data)
Congratulations for reaching lesson 2. Welcome to this lesson on
Calculating measures of position for grouped data. Have fun!
After going through this module, you are expected to:
Calculate quartile, decile, percentile in grouped data.
PRE-ASSESSMENT
Activity 2: Remember Me!
To check your readiness, review the previous lesson by answering this
activity 2.
The grade 10 learner having their exercise during their MAPEH period
before weighing. Ten of them have their weights 75, 65. 70, 85, 72, 80, 67,
58, 77, and 65.
Calculate the following quantiles:
2.)
3.)
4.)
5.)
You had learned and familiarized the steps on how to calculate the
measures of Position for ungrouped data in the previous lesson? In this
lesson, the data will be in a large scale set of data. It is also divided in
several parts equally by four, ten, and a hundred called quartile, decile, and
percentile respectively. They are called the Quantiles. Quantiles are in the
Measures of Position.
PRESENTATION: Grouped data
Grouped data are dealing with the large amount of data from any group
of an experiment. It is used to determine the huge scales data. For example
the National data.
Measures of Positions for Grouped data is used by the government or
business companies in their study on how the income is distributed in the
country or company. The best example is How much is the total income
earned by low wage earning groups and by high wage earning groups? If
both groups earn with same proportion, then there is an income equality.
14
CONCEPT DEVELOPMENT
The Quartile for Grouped Data
Remember that quartiles divide the distribution into four equal parts.
In getting the median of the measures of central tendency is similar in
calculating the
and
. In getting the median we first get the median
class. And in getting the
and
we must find first the class before
computing its values.
The first quartile
class is the class interval where the {
is contained, and the third quartile
class interval is {
}th score
}th score.
The formula in computing the Quartiles of Grouped Data is
= LB + {
}í
Where:
= 1st quartile,
LB = lower boundary of the
= 2nd quartile, and
= 3rd quartile,
class,
N = total frequency,
í = size of the class interval,
= frequency of the
= cumulative frequency of the class before the
k
class,
class,
and
= nth quartile, where n= 1, 2, and 3
Lower Boundaries (LB) are the lowest scores in each class which is less
by o.5. Example, the LB of the class 21-25. 21 is the smallest boundary,
then 21 minus 0.5 is 20.5. or 21 – 0.5 = 20.5
Cumulative Frequencies (cf) are the frequency which is added by
frequency in each class going up and/or total frequency from the top less
by each frequency going down.
Scores
Frequency
46-50
4
41-45
6
36-40
10
31-35
8
26-30
12
21-25
5
i=5
∑=45
15
cf
This is how to determine the cumulative frequency. Determine first the
total number of frequency. If you have total frequency, then follow the
column on going down. You can still use the going up column by copying
the frequency of the lower class then follow the steps below.
Scores
Frequency
Going up
cf
Going down
46-50
4
41+4
45
N=45
41-45
6
35+6
41
45-4
36-40
10
25+10
35
41-6
31-35
8
17+8
25
35-10
26-30
12
5+12
17
25-8
21-25
5
5
5
17-12
i=5
∑=45
For example.
Calculate the
and
of the Mathematics 9 quiz results of 45
learners as presented in the table below.
Scores
Frequency
46-50
4
41-45
6
36-40
10
31-35
8
26-30
12
21-25
5
Solutions:
Scores
Frequency
LB
Less than (<cf)
46-50
4
45.5
45
41-45
6
40.5
41
36-40
10
35.5
35
31-35
8
30.5
25
26-30
12
25.5
17
21-25
5
20.5
5
i= 5
∑= 45
16
To determine the size of the class interval (i) count the number of scores
in each class. Example for class 21-25 the size is 5 since it has 5 scores; 21,
22, 23, 24, and 25.
To determine the sum of the frequency, just add all frequencies.

class:
=
=
=11.25
This means we will locate the class interval which contain the 11.25th
score. It happens that the 5th to17th score was in the class 26-30. Then
class is in class 26-30.
So,
LB = 25.5,
N = 45,
= LB + {
= 5,
= 12,
}í = 25.5 + {
í= 5, and k = 1
}5 = 25.5 +2.60 = 28.10
= 28.10
To find
=
…..
=
=
= 22.5, then the class was in 31-35 since the 18th -25th score
was in this class. Since,
is the formula on how to find the class of
So,
LB = 30.5,
= LB + {
N = 45,
}í = 30.5 + {
= 17,
= 8,
í= 5 and k = 2
}5 = 30.5 + 3.4375 = 33.94
= 33.94
17
To find
=
….
=
=
= 33.75, then the class was in 36-40 since the 26th -35th
score was in this class. Since,
is the formula on how to find the class of
.
So,
LB = 35.5,
N = 45,
= 10,
= LB + {
= 25,
í= 5, and
}í = 35.5 + {
k=3
}5 = 35.5 + 0.875 = 36.375
= 36.375,
To check your understanding, Try Activity 2.1.
The Decile for Grouped Data
You have learned about quartile for grouped data in the previous pages.
In this topic, you will learn about decile for grouped data.
The decile are the values divided a score into ten equal parts. Denoted as
, , , , , , , , . They are computed in the same way with the
quartiles are calculated.
The formula in computing the Deciles of Grouped Data is
= LB + {
}í
Where:
= 1st decile,
N = total frequency,
= 2nd decile,
= 9th decile,
LB = lower boundary of the
= cumulative frequency of the class before the
= frequency of the
k
class,
class
class
í= size of the class interval
= nth quartile, where n= 1, 2, 3, 4, 5, 6, 7, 8, and 9.
18
Study the example below.
Calculate the
learners.
and
of the Mathematics 9 quiz results of 45
Scores
Frequency
46-50
4
41-45
6
36-40
10
31-35
8
26-30
12
21-25
5
Solutions:
How can you solve this problem? Study the given example. Follow the
steps on how to determine the lower boundary and the less cumulative
frequency for a previous topic.
Scores
Frequency
LB
Less than (<cf)
46-50
4
45.5
45
41-45
6
40.5
41
36-40
10
35.5
35
31-35
8
30.5
25
26-30
12
25.5
17
21-25
5
20.5
5
i= 5
∑= 45
To find
……
=
=
=
= 13.5, then the class was in 26-30 since the 6th- 17th score
was in this class. Since,
LB = 25.5,
= LB + {
is the formula on how to find the class of
N= 45,
= 5,
}í=25.5 + {
= 12, í= 5,
and k = 3
}5 = 25.5 + 3.54 = 29.04
= 29.04
19
.
To find
=
=
……
=
= 31.5, then the class was in 36-40 since the 26th- 35th score
was in this class. Since,
LB = 35.5,
= LB + {
is the formula on how to find the class of
N= 45,
= 25,
}í=35.5 + {
= 10,
.
í = 5, and k = 7
}5 = 35.5 + 3.25 = 38.75
= 38.75
The percentile for Grouped Data
The percentile of grouped data are determined according to the
percentage below their values. The percentile for grouped data is similar to
that of finding quartiles and deciles for grouped data. The kth percentile
denoted by
is computed as follows:
= LB +
{
}í
Where:
= unknown percentile,
N = total frequency,
LB = lower boundaries
í = class interval,
= frequency of the class,
= cumulative frequency before the class,
k = nth decile
For example:
Calculate the
learners.
and
of the Mathematics 9 quiz results of 45
Scores
Frequency
46-50
4
41-45
6
36-40
10
31-35
8
26-30
12
21-25
5
20
Solutions:
Follow the steps on how to determine the lower boundary and the less
cumulative frequency from the previous topic.
Scores
Frequency
LB
Less than (<cf)
46-50
4
45.5
45
41-45
6
40.5
41
36-40
10
35.5
35
31-35
8
30.5
25
26-30
12
25.5
17
21-25
5
20.5
5
i= 5
∑= 45
To find
…..
=
=
=
= 20.25, then the class was in 31-35 since the
18th - 25th score was in this class. Since,
the class of
.
LB = 30.5,
N = 45,
= LB + {
To find
=
is the formula on how to find
}í
= 17,
= 30.5 + {
= 8,
í = 5, and k = 45
}5 = 30.5 + 2.03 ,
= 32.5
……
=
= 40.50, then the class was in 41-45 since the 36th-
=
41th score was in this class. Since,
class of
is the formula on how to find the
.
LB = 40.5,
= LB + {
N= 45,
}í ,,,,
= 6,
= 35,
= 40.5 + {
í= 5, and
k = 45
}5 = 40.5 + 4.58 = 40.08
= 40.08
If you have no questions on this topic, do Activity 2.4 Look for Me!
21
ACTIVITIES : Activity 2.4: Look for Me!
Joenin was a scholar in an Institution which requires a grade above 9
percent. In what class Interval should Joenin belong in order for him to
maintain his scholarship? Find the value of the ,
,
,
,
, and
,
for the given distribution below.
Class Interval(Score)
frequency
96-100
8
91-95
11
86-90
6
81-85
9
76-80
7
71-75
4
lb
<cf
ASSESSMENT
Joenin belongs to the Science class of the Grade 10 at San Andres NHSCabadiangan Annex. All of them are Achievers and Honors Student. He
claims to be on TOP. Where can you find Joenin base on the data given?
Find the following quantiles based from the given data in the table.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
91.1
F
Class Interval(Grades)
93.7
E
Frequency(f)
97-99
1
94-96
3
91-93
14
88-90
12
85-87
8
89.3 92.32 89.11 93.22 87.875
R
S
T
_ _ _ _ _
I
_ _ _ _ _
22
P
90.25 92.73
C
A
91.83
L
GENERALIZATION/SYNTHESIS

The quartiles are the score points which divide a distribution into
four equal parts. Twenty-five (25%) per cent of the distribution are
below the first quartile, fifty (50%) per cent are below the second
quartile, and seventy-five (75%) per cent are below the third quartile.
 The decile is the nine score points which divide a distribution into ten
equal parts.
 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 according to
percentage below their values.
 Grouped data are dealing with the large amount of data from any
group of an experiment. It is used to determine the huge scales data.
 Ungrouped data is a set of data taken from a small group of people
or any group of an experiment. It is taken from an individual from a
very small group.
 To calculate the Measures of Position through the following formula:
quartile, decile, and percentile, the following formulae are used.
Quartile:
Decile:
Percentile:

=
=
=
To calculate the Measures of Position such as quartiles, deciles, and
percentiles the following formulae are used.
Quartile:
= LB + {
}í
Decile:
= LB + {
}í
Percentile:
= LB + {
}í
Quantiles are measures of Position such as quartile, decile and
percentile.
23
APPLICATION
Make your own frequency distribution. Gather data from your
classmate’s General average. Then create a table for grouped data and
calculate the unknown quantiles below.
Score
Frequency Cumulative Lower
Frequency Boundaries
i=
∑N =
Find the Following Quantiles:
a.)1st quartile
b.) 85th percentile
c.)3rd quartile
d.)4th decile
e.)6th decile
f.)45th percentile
Post-Assessment
Choose the letter of the correct answer. Write the chosen letter on a
separate sheet of paper.
1. Find the 30th Percentile In a set of scores: 9, 16, 12, 13, 13, 20, 11,
21, and 18.
A. 9
B. 12
C. 13
D. 16
2. Find the 90th Percentile in the data above
A. 13
B. 16
C. 20
D. 21
3. Find the Lower quartile of the following data listed. 85, 100, 79, 93,
86, 76, 90, 49, 109, 59, 39, and 69.
A.49
B. 59
C.69
D. 76
24
4. What is the Fifth Decile of the age of a grade 10 learners whose ages
are 15, 18, 17, 16, and 15?
A.15
B. 16
C.17
D. 18
5. In a set of scores: 9, 16, 12, 13, 13, 20, 11, 21, and 18. The
is _____.
A.16
B. 18
C.20
D. 21
6. In a set of scores: 9, 16, 12, 13, 13, 20, 11, 21, and 18, find
A. 13
B. 16
C. 18
D. 20
For items 9 to 15, refer to Table below.
Score
46-50
41-45
36-40
31-35
26-30
21-25
i= 5
7.
Find the
A. 31.48
C. 33.48
Frequency Cumulative Lower
Frequency Boundaries
2
45
45.5
4
43
40.5
12
39
35.5
14
27
30.5
10
13
25.5
3
3
20.5
∑N = 45
of the Mathematics test results of 45 students.
B. 32.48
D. 34.48
10.What cumulative frequency must be used in solving for
A. 3
B. 13
C. 27
D. 39
11.The eight decile is ___________
A. 36.52
B. 38.25
C. 38.52
D. 39.25
13.The third quartile is _______________
A. 36.31
B. 37.31
C. 38.31
D. 39.31
25
?
.
Answer Key
Preassessment
page
Pre-assessment
lesson 1
The median is 8
1.
2.
3.
4.
5.
6.
7.
b
a
b
b
d
b
b
Application lesson 1
Cross Quantile
Puzzle
Across
1.483
Activity
lesson 1
Assessment
lesson 2
FIRST PLACE
Post-Assessment
Lesson 2
1. b
2. d
3. b
4. b
5. c
6. d
7. a
8.b
9.d
10.c
2.463
4.525
6.323
Pre-assessment
Lesson 2
= 7.7≈8=77
2.)
= 6.6≈7=75
3.)
= 8.8≈9=80
4.)
= 2.75≈3=65
5.)
=7.15≈7=75
Activities lesson 2.4
Look for Me
= 87.58
=2
7.483
=6
Down
= 12/5
1.454
= 28/5
3.358
= 85.63
= 14/5
5.553
= 92.38
= 16/5
8.248
assessment Lesson 1
= 77
= 65
= 81.61
= 95.045
= 97.68
Application lesson 2
= 80
= 67
= 77
= 80
Answers may vary
26
Reference/s:
*Emmanuel P. Abuzo et. Al., Mathematics 8 Learner’s Module, First Edition
2013, ISBN, Book Media Press Inc.
*Mathematics 8 Teacher’s Guide
*Melvin M. Callanta et. Al., Mathematics-Grade 10 Learning Module First
Edition 2015, Department of Education, Rex Bookstore, Inc.
Website Links as references and Sources of Learning
Activities:
http://www.youtube.com/watch?v=uYII2M9YwHE&feature=share
This videos provides formula, examples and exercises of Measures of position for
Ungrouped data in Tagalog version.
http://www.youtube.com/watch?v=GBWS5iPTNDC&feature=share
This videos provides formula, examples and exercises of decile for grouped data.
http://youtu.be/30NPKaIHE1w
This videos provides formula, examples and exercises of quartiles for grouped data.
http://www.youtube.com/watch?v=fKI73nBOBsE&feature
This videos provides formula, examples and exercises of Measures of position for
grouped data.
27
For inquiries and feedback, please write or call:
Department of Education –Learning Resources Management and
Development Center(LRMDC)
DepEd Division of Bukidnon
Fortich Street, Sumpong, Malaybalay City, Bukidnon
Telefax:
((08822)855-0048
E-mail Address:
bukidnon@deped.gov.ph
28