MATHEMATICS 10

NAME : ________________________________________GRADE AND SECTION : _______________

TEACHER : _________________________ QUARTER 4 – WEEK 1 SCORE : ________

ILLUSTRATING QUARTILES, DECILES, AND PERCENTILES

Background Information
Measures of Position are important tools in statistics. They tell us
where a particular data value falls in the distribution compared to the data as a
whole.
The most common measures of position are quartiles, deciles and
percentiles.

QUARTILES

A quartile divides a sorted data set into 4 equal parts, so that each part
represents ¼ or 25% of the data set.
They are following: first quartile, second quartile, and third quartile. They are
denoted by Q1 , Q2, and Q3 respectively.
The difference between Q3 and Q1 is the interquartile range.

25% 25% 25% 25%

25% of all the data has a value ≤ Q1
50% of all the data has a value ≤ M or Q2
75% of all the data has a value ≤ Q3
Lower M Upper 50% of all the data lies between Q1 and Q3
Quartile Median Quartile
Q1 Q3
Q2

FORMULAS IN GETTING QUARTILES

𝟏 𝟏 𝟑
𝑸𝟏 = (𝒏+𝟏) 𝑸𝟐 = (𝒏+𝟏) 𝑸𝟑 = (𝒏+𝟏)
𝟒 𝟐 𝟒

Note : n is the total number of items in the distribution

 If the value of Q1 , Q2 , and Q3 is decimal, round off to the nearest
integer .

Illustrative Example # 1 – The scores of seven (7) randomly selected

students in a 20-item Mathematics Quiz are as follows:
13 14 15 17 18 19 20
Q1 Q2 Q3

a. What is the lower quartile?

b. Determine the second quartile.
c. What is the upper quartile?
d. Calculate the interquartile range.

Solutions :
a. lower quartile
The lower quartile is Q1 and Q1 = 14.
This means that 25% of the students scored less than or equal to 14 in the 20-
item Math Quiz.
b. second quartile
Q2 is the second quartile , Q2 = 17
This means that 50 % of the students scored less than or equal to 17.
c. upper quartile
Q3 is the upper quartile , Q3 = 19
This means that 75% of the students scored less than or equal to 19.
We can also say that 25% of the students scored greater than or equal to 19 in
the 20-item Math Quiz.
d. Interquartile range
The interquartile range is the difference between the upper and lower
quartile.
Interquartile range = Q3 - Q1
= 19 - 14
= 5

Illustrative Example # 2 – During the month of February this year,

Dr. M. Rigor recorded the number of COVID-19 recovery patients who came out to
the hospital each day. The results are 15, 13, 10, 14, 21, 18, 16, 9, 14, 8 and 15.
Find the : a. lower quartile
b. middle quartile
c. upper quartile
d. Interquartile range
Solutions :
Arrange first the given data in ascending / increasing order.
8, 9, 10, 13, 14, 14, 15, 15, 16, 18, 21

Least value
Q1 Q2 Middle value
Q3
Greatest value

a. Lower quartile
The lower quartile ( Q1 ) is the value that is between the middle value and the
least value in the set of data. In our given set of data, the lower quartile
( Q1 ) is 10.
b. Middle quartile
The middle quartile is also the middle value in the given set of data. In our
example, the middle quartile ( Q2 ) is 14 .
c. Upper quartile
The upper quartile ( Q3 ) is the value that is between the middle value and the
greatest value in the data. Therefore, the upper quartile is 16.
d. Interquartile range
Interquartile range = Q3 - Q1
= 16 - 10
= 6

Another solution ( using the formulas for quartile )

Arrange first the given data in ascending / increasing order.
8, 9, 10, 13, 14, 14, 15, 15, 16, 18, 21
n = 11
a. Lower quartile
𝟏 𝟏 𝟏
𝑸𝟏 = (𝒏+𝟏) = ( 𝟏𝟏 + 𝟏 ) = ( 𝟏𝟐 ) = 3
𝟒 𝟒 𝟒

This means that the lower quartile is the 3rd data element. 10 is the 3rd element in our data set,
therefore Q1 = 10

b. Middle quartile
𝟏 𝟏 𝟏
𝑸𝟐 = (𝒏+𝟏) = ( 𝟏𝟏 + 𝟏 ) = ( 𝟏𝟐 ) = 6
𝟐 𝟐 𝟐

This means that the lower quartile is the 6th data element. 14 is the 6th element in our data set,
therefore Q2 = 14
c. Upper quartile
𝟑 𝟑 𝟑
𝑸𝟑 = (𝒏+𝟏) = ( 𝟏𝟏 + 𝟏 ) = ( 𝟏𝟐 ) = 9
𝟒 𝟒 𝟒

This means that the lower quartile is the 9th data element. 16 is the 9th element in our data set,
therefore Q3 = 16

DECILES
Deciles are the nine score points which divide a distribution into ten equal parts.
They are the following: first decile, second decile, third decile, and so on and are
denoted by D1, D2, D3,…D9.

10% of all the data has a value ≤ D1 60% of all the data has a value ≤ D6
20% of all the data has a value ≤ D2 70% of all the data has a value ≤ D7
30% of all the data has a value ≤ D3 80% of all the data has a value ≤ D8
40% of all the data has a value ≤ D4 90% of all the data has a value ≤ D9
50% of all the data has a value ≤ D5

FORMULA IN GETTING DECILE

𝒌 n = total number of observations
𝑫𝒌 = (𝒏+𝟏)
𝟏𝟎
k = decile rank

Illustrative Example # 3 – The amounts of profit (in Pesos) of nineteen

(19) young entrepreneurs in their School Bazaar Day in the morning are shown below.
23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 60
a. What is the 2nd decile?
b. Find the 8th decile.
c. What is the 5th decile?
d. How many entrepreneurs are above the 6th decile?
Solutions :
a. 2nd decile :
Given : n= 19 k=2
𝒌 𝟐 𝟐
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = ( 𝟏𝟗 + 𝟏 ) = (𝟐𝟎) = 4
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 2nd decile is the 4th data element in the given set of
data. Therefore, D2 = 30.
b. 8th decile
Given : n = 19 k= 8
𝒌 𝟖 𝟖
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = ( 𝟏𝟗 + 𝟏 ) = (𝟐𝟎) = 16
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 8th decile rank is the 16th data element in our given set
of data.
23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 60
The 8th decile in our given set of data is 55.
c. 5th decile
Given : n = 19 k= 5
𝒌 𝟓 𝟓
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = ( 𝟏𝟗 + 𝟏 ) = (𝟐𝟎) = 10
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 5th decile rank is the 10th data element in our given set
of data.
23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 60
The 5th decile in our given set of data is 36.
d. How many entrepreneurs are above the 6th decile?
To answer the question , we will first locate the 6th decile in our given set of
data.
Given : n = 19 k= 6
𝒌 𝟔 𝟔
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = ( 𝟏𝟗 + 𝟏 ) = (𝟐𝟎) = 12
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 6th decile rank is the 12th data element in our given set
of data.
23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 60
Since our 6th decile number is 45, therefore there are 7 entrepreneurs that are
above the 6th decile.

Illustrative Example # 4 – Damdeok wants to donate for the VNHS

Pilot Implementation of Limited Face to Face Classes. He sells facemasks online
and saves the profit for donation. The following data set is a list of his profit in peso
for nine days: 150, 208, 230,256, 275, 245, 196, 300 , 320.
Find the value of the 5th, 7th and 9th decile.
Solutions : Arrange first the set of data in ascending order
150 , 196, 208, 230, 245, 256, 275, 300 , 320
a. 5th decile
n=9 k = 5
𝒌 𝟓 𝟓
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = (𝟗+𝟏) = (𝟏𝟎) = 5
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 5th decile rank is the 5th data element in our given set of data.
150 , 196, 208, 230, 245, 256, 275, 300 , 320
Therefore D5 = 245.

b. 7th decile
n=9 k = 7
𝒌 𝟕 𝟕
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = (𝟗+𝟏) = (𝟏𝟎) = 7
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 7th decile rank is the 7th data element in our given set of data.
150 , 196, 208, 230, 245, 256, 275, 300 , 320
Therefore D7 = 275.

c. 9th decile
n=9 k = 9
𝒌 𝟗 𝟗
𝑫𝒌 = (𝒏+𝟏) 𝑫𝟐 = (𝟗+𝟏) = (𝟏𝟎) = 9
𝟏𝟎 𝟏𝟎 𝟏𝟎

This indicates that the 7th decile rank is the 7th data element in our given set of data.
150 , 196, 208, 230, 245, 256, 275, 300 , 320
Therefore D9 = 320.

PERCENTILES

Percentiles are the ninety-nine score points which divide a distribution into
one hundred equal parts, so that each part represents the data set. The first
percentile, second percentile, third percentile and so on up to the ninety-ninth
percentile are denoted by P1, P2, P3, … P97, P98, P99.
P1 separates the lowest 1% from the other 99%
P2 separates the lowest 2% from the other 98%
P3 separates the lowest 3% from the other 97%
.
 .
P99 separates the lowest 99% from the other 1%
FORMULA IN GETTING PERCENTILES
𝒌 n = total number of observations
𝒑𝒌 = (𝒏+𝟏)
𝟏𝟎𝟎
k = percentile rank

Illustrative Example # 5 – Nineteen (19) randomly selected family members

in Barangay San Gavino are interviewed by STEM Grade 12 learners for a research
study. Their present ages are shown below.
13, 14, 15, 15, 16, 18, 19, 20, 22, 22, 22, 23, 24, 25, 26, 27, 27, 29, 30
a. What is the 30th percentile?
b. Find the 90th percentile.
c. What age is at the middle?
d. How many family members are below 25% of the data?
e. How many family members are above 75% of the data?

Solutions :
a. 30th percentile Given : n = 19 k = 30
𝒌 𝟑𝟎 𝟑𝟎
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟗 + 𝟏 ) = ( 𝟐𝟎 ) = 6
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

13, 14, 15, 15, 16, 18, 19, 20, 22, 22, 22, 23, 24, 25, 26, 27, 27, 29, 30
P30 = 18. This means that 30 % of the family members being interviewed are
below or equal to 18 years old.

b. 90th percentile Given : n = 19 k = 90

𝒌 𝟗𝟎 𝟗𝟎
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟗 + 𝟏 ) = ( 𝟐𝟎 ) = 18
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

13, 14, 15, 15, 16, 18, 19, 20, 22, 22, 22, 23, 24, 25, 26, 27, 27, 29, 30
P90 = 29. This means that 90 % of the family members being interviewed are
below or equal to 29 years old.

c. Middle age is the 50th percentile Given : n = 19 k = 50

𝒌 𝟓𝟎 𝟓𝟎
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟗 + 𝟏 ) = ( 𝟐𝟎 ) = 10
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

13, 14, 15, 15, 16, 18, 19, 20, 22, 22, 22, 23, 24, 25, 26, 27, 27, 29, 30
 P50 = 22. This means that 50 % of the family members being interviewed are
below or equal to 22 years old.

d. family members below 25 % of the data

𝒌 𝟐𝟓 𝟐𝟓
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟗 + 𝟏 ) = ( 𝟐𝟎 ) = 5
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

13, 14, 15, 15, 16, 18, 19, 20, 22, 22, 22, 23, 24, 25, 26, 27, 27, 29, 30
The 25th percentile is 16 and the numbers below 16 are 13, 14 , 15 and 15 .
Therefore, there are 4 family members in Barangay San Gavino that are below the
25th percentile.

e. family members above the 75 % of the data

𝒌 𝟕𝟓 𝟕𝟓
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟗 + 𝟏 ) = ( 𝟐𝟎 ) = 15
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

13, 14, 15, 15, 16, 18, 19, 20, 22, 22, 22, 23, 24, 25, 26, 27, 27, 29, 30
The 75th percentile is 26 and the numbers greater than 26 are 27, 27,29 and
30 . Therefore, there are 4 family members in Barangay San Gavino that are above
the 75th percentile

Illustrative Example # 6 – In celebration of the Founding Anniversary

of the town of Victoria, a fund raising marathon was conducted. The data below
are the distances travelled (in kilometers) by fifteen (15) participants .

5 6 6 7 8 9 10 11 12 13 14 15 16 17 17
a. What is the 25th percentile?
b. Determine the 50th percentile.
c. What is the 75th percentile?

Solutions :
a. 25th percentile Given : n = 15 k = 25
𝒌 𝟐𝟓 𝟐𝟓
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟓 + 𝟏 ) = ( 𝟏𝟔 ) = 4
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

5 6 6 7 8 9 10 11 12 13 14 15 16 17 17

P25 = 7 . This denotes that 25% of the participants travelled less than or equal to 7
kilometers
b. 50th percentile Given : n = 15 k = 50
𝒌 𝟓𝟎 𝟓𝟎
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟓 + 𝟏 ) = ( 𝟏𝟔 ) = 8
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

5 6 6 7 8 9 10 11 12 13 14 15 16 17 17

P50 = 8 . This denotes that 50% of the participants travelled less than or equal to
11 kilometers

c. 75th percentile Given : n = 15 k = 75

𝒌 𝟕𝟓 𝟕𝟓
𝒑𝒌 = (𝒏+𝟏) = ( 𝟏𝟓 + 𝟏 ) = ( 𝟏𝟔 ) = 12
𝟏𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎𝟎

5 6 6 7 8 9 10 11 12 13 14 15 16 17 17

P75 = 15 . This denotes that 75% of the participants travelled less than or equal to
15 kilometers

For further understanding of the lesson, visit the link below:

https://youtu.be/y5iAygQLVeg

LEARNING COMPETENCY WITH CODE

M10SP- Iva – 1 - Illustrate the following measure of position:
quartiles, deciles and percentiles

ACTIVITY 1 - Read each item carefully and choose the letter of the
correct answer. Write the letter of your answer on a
separate sheet of paper. CAPITAL LETTERS
ONLY.
1. When a distribution is divided into four equal parts, each score point that
describes the distribution is called a __________.
A. quartile B. decile C. percentile D. mean
2. Deciles are the score points that divide the distribution into how many equal
parts?
A. two B. four C. ten D. hundred
3. The following are equivalent to 50% EXCEPT?
A. D5 B. Q2 C. P50 D. D50
4. In a 50- item test, you got 35 which is the 75th percentile. What does it mean?
A. You got the highest score.
B. Your score is higher than 25 % of your classmates.
C. 25 % of your classmates got the scores higher than 35.
D. 75 % of your classmates did not pass the test.

5. Find D2 in the given set of data: 7, 10, 11, 15, 18, 21, 25, 29, 31
A. 7 B. 10 C. 11 D. 15

6. Interquartile range is the difference between __________.

A. Q2 and Q1 C. Q3 and Q1
B. Q4 and Q3 D. Q3 and Q1

7. What is the Q1 in the set 44, 21, 59, 64, 74, 57, 69, 38, 40, 27, and 81?
A. 27 B. 38 C. 40 D. 44

8. Which is equivalent to the fifth decile?

A. median B. third quartile C. first quartile D. interquartile range

9. If Q1 = 15 and Q3 = 23, then the interquartile range is

A. 7 B. 8 C. 9 D. 10

10. If the passing score of a 100-item standardized test is 50 and considered as the
60th percentile, which one from the following statements is NOT TRUE?
A. 60% of the examiners got a score lower than or equal to 50.
B. 40% of the examiners garnered a score of 50 and above.
C. Obtaining a score that is in the 5th decile means passing the test.
D. 40% of the examiners answered 50 or more questions correctly.

ACTIVITY 2 - Solve for the indicated quartile position. Write your

complete solution in your answer sheet.
Below are the scores of selected Grade 10 learners in a Mathematics Activity.
20, 24, 28, 30, 22, 38, 45
Solve for the : a. lower quartile ( Q1 )
b. middle quartile ( Q2 )
c. upper quartile ( Q3 )
 ACTIVITY 3 - Refer to the given situation below and find the value of
the indicated measure of position. Write your complete solution
and answer on a sheet of paper.

The owner of MaarTEA , a milk tea shop recorded the number of customers
who came into his café each hour a day. The results were as follows :

12, 13, 14, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32

Determine the indicated measures of position below :

1. P25 6. Q3
2. D8 7. P50
3. D4 8. P10
4. Middle quartile 9. Lower quartile
5. Interquartile 10. P60

REFLECTION / JOURNAL WRITING

What is the importance of different roles or positions in an organization?

Express your idea in at least 10 – 15 sentences.

REFERENCES : Quarter 4 – Week 1 Learning Activity Sheet ( SDO Tarlac City )

Quarter 4 – Module 4 – Quartile for Ungrouped Data ( SDO Pasig City )
Quarter 4 – Module 5 – Decile for Ungrouped Data ( SDO Pasig City )
Quarter 4 – Module 6 – Percentile for Ungrouped Data ( SDO Pasig City )
Quarter 4 – Module 1 – Illustrating Quartiles, Deciles and Percentiles ( DepED CO )