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MATHEMATICS
Quarter 4 – Module 1
Illustrating The Measures of Position
for Ungrouped Data
Mathematics – Grade 10
Alternative Delivery Mode
Quarter 4 – Module 1: Illustrating the following Measures of Position for
Ungrouped Data
First Edition, 2020

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Mathematics
Quarter 4 – Module 1
Illustrating the Measures of
Position for Ungrouped Data
 I

LEARNING COMPETENCY:
• Illustrate the following measures of position: quartiles,
deciles, and percentiles. (M10SP - IVa - 1)
OBJECTIVES:
K: Define the measures of position: quartiles, deciles, and
percentiles.
S : Illustrate quartiles, deciles, and percentiles.
: Calculate specified measure of position for ungrouped
data.
A: Perform the given task with accuracy.

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

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 Lesson Illustrating the Following Measures of
Position: Quartiles, Deciles and Percentiles
and percentiles.

Ezra Louis- March 1, 2018

Did you take the National Career Assessment Examination (NCAE) when you were in
Grade 9? If so, what was your score? Did you know your rank?

Have you thought of comparing your academic performance with that of your
classmates?

Have you wondered what score you need for each subject area to qualify for honors?
Have you asked yourself why a certain examinee in any national examination gets
higher rank than the other examinees?

Some state colleges and universities are offering scholarship programs for graduating
students who belong to the upper 5%, 10%, or even 25%. What does this mean to you?

In this module, you will study about the following measures of position: quartiles,
deciles, and percentiles. Be ready to answer the following questions:
1. How would I know my position given the academic rank?
2. What are the ways to determine the measure of position in a set of data?

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 ’s In

Let us review the

Find the median of the following set of data. concept of median,
which is one of the
1. 4, 7, 1, 8, 10 concepts needed in the
2. 8, 6, 6, 10, 85 study of this module.

3. 34, 43, 45, 1, 30, 4

4. 29, 3, 42, 17, 17, 48, 7
5. 45, 47, 2, 44, 42, 27

How did you find the median of the set of data?

➢ To find the median:
1. Arrange the numbers from smallest to largest.
2. The number in the middle is the median. If there are two middle numbers, add them
and divide by two.

The median divides the distribution into two equal parts. It is a point of distribution
where one-half of the distribution lies below it or above it.

’s New

What’s the meaning of this?

Write your initial definition of the different measures of position.

Measures of Positions My Initial Definition

1. Quartile

2. Decile

3. Percentile

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 is It

Discussion:
➢ How do you define Measures of Position?
Answer: They are techniques that divide a set of data into equal groups.
➢ What are the different measures of position?
Answers: Quartiles, Deciles, Percentiles
➢ Can you define Quartiles?
Quartiles are points that divide the ranked data into four equal parts. Each set of
data has three quartiles.

25% 25% 25% 25%

L Q1 Q2 Q3 H
L = lowest score
Q1 = First quartile or lower quartile
Q2 = second quartile or middle quartile
Q3 = third quartile or upper quartile
H = highest score

1.First quartile (Q1) is the value in the data set such that 25% of the data points are less than
this value and 75% of the data set is greater than this value.
2. Second quartile (Q2) is the value in the data set such that 50% of the data points are less
than this value and 50% of the data set are greater than this value.
3. Third quartile (Q3) is the value such that 75% of the values are less than this value and 25%
of the values are greater than this value.
4. Interquartile range is the difference between the upper quartile (Q3) and the lower
quartile (Q1) in a set of data.

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The Quartile for Ungrouped Data
Illustrative examples
Example 1. A group of students obtained the following scores in their statistics quiz: 8 , 2 , 5 ,
4,8,5,7,1,3,6,9

Solution:

Observe how the lower quartile (Q1), middle quartile (Q2), and upper quartile (Q3) of the
scores are obtained. Complete the statements below:
➢ The first quartile 3 is obtained by observing the position of 3 which is in the middle of
the numbers from 1 to 5. (observe the position of 3 from 1 to 5)
➢ The second quartile, 5, is obtained by observing the position of 5 which is in the
middle of the numbers from 1 to 9. (observe the position of 5 from 1 to 9)
➢ The third quartile, 8, is obtained by observing the position of 8 which is in the middle
of the numbers from 6 to 9. (observe the position of 8 from 6 to 9).

Mendenhall and Sincich Method

Using Statistics for Engineering and the Sciences, define a different method of finding
quartile values.
To apply their method on a data set with n elements, first calculate:
1
Lower Quartile (L) = Position of Q1 = (n + 1)
4
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:
3
Upper Quartile (U) = Position of Q3 = 4 (n + 1)
and round to the nearest integer. If U falls halfway between two integers, round down. The
Uth element is the upper quartile value (Q3).

Example 2. Example data set: {1, 3, 7, 7, 16, 21, 27, 30, 31} and n = 9.
1
To find Q1, locate its position using the formula Q1 = 4 (n + 1)
and round off to the nearest integer.
1
Position of Q1 = 4 (n + 1)
1
= (9 + 1)
4
1
= (10)
4
= 2.5
The computed value 2.5 becomes 3 after rounding up. The lower quartile value (Q 1) is the 3rd
data element, so Q1 = 7.

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 1, 3, 7, 7, 16 , 21, 27, 30 , 31
3
Position of Q3 = (n + 1)
4
3
= (9 + 1)
4
3 30
= (10) = = 7.5
4 4

The computed value 7.5 becomes 7 after rounding down. The upper quartile value (Q 3) is the
7th data element, so Q3 = 27

1, 3, 7, 7, 16 , 21, 27, 30 , 31

Example 3.
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
b. Second, locate the position of the score in the distribution.
𝟏
Position of Q1 = 𝟒 ( n + 1 )
𝟏
= (9+1)
𝟒
𝟏
= 𝟒 ( 10 )
= 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.

d. Find the upper quartile

𝟑
Position of Q3 = 𝟒 ( n + 1 )
𝟑
= (9+1)
𝟒
𝟑
= 𝟒 ( 10 )
= 7.5

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).
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 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

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.

Illustrative example
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 n = 10
Second, find decile value on a data with n elements:
3
To find its D3 position, use the formula (n + 1) and round off to the nearest integer
10
3
Position of D3 = (10 + 1)
10
3
= (11)
10
33
= 10 = 3.3 = 3
D3 is the 3rd element. Therefore, D3 = 20.

The Percentile for Ungrouped Data

The percentiles are the ninety-nine score points which divide a distribution into one hundred
equal parts, so that each part represents the data set. They are used to characterize values
according to the percentage below them. For example, the first percentile (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 P 30. The 1st decile is the
10th percentile (P10 ). It means 10% of the data is less than or equal to the value of P 10 or D1,
and so on.

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Illustrative example:
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 n = 10 , k = 30

Solution:
Arrange the scores from the lowest to the highest.
15 18 20 23 28 28 33 35 40 42

Steps to find percentile value on a data with n elements:

𝑘 (𝑛+1)
To find its P30 position use the formula 100 n and round off to the nearest integer.
30 (10+1) 30 (11) 330
Position of P30 = = = = 3.3 ≈ 3
100 100 100

P30 is the 3rd element. Therefore, P30 = 20

15 18 20 23 28 28 33 35 40 42

’s More
ACTIVITY:

Solve the following problems:

1. Albert has an assignment to randomly ask 10 students in their school about their ages. The
data are given in the table below.
Name Age Name Age
Ana 10 Tony 11
Ira 13 Lito 14
Susan 14 Christian 13
Antonette 13 Michael 15
Gladys 15 Dennis 12

a. What is Q1, Q2, and Q3 of their ages?

b. How many students belong to Q1, Q2, and Q3 in terms of their ages?
2. Mrs. Lim gave a test to her students in Statistics. The students finished their test in 35
minutes. This time is the 2.5th decile of the allotted time. What does this mean?

3. Given 30 enumeration and multiple-choice items in their summative test in Mathematics.

a. Find the 75th percentile or P75 of the following test scores of 15 students from Grade
10-A
22, 10, 13, 13, 20, 13, 11, 17, 18, 14, 21, 15, 21, 16, 15
b. How many students have passed the exam?

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 I Have Learned

Directions:
Write True if the statement is true for any given data set; otherwise, write False.
_____ 1. The 5th decile is equal to the 2nd quartile.
_____ 2. The mean is equal to the 2nd quartile.
_____ 3. P50 is equal to the median.
_____ 4. Q3 is equal to P75.
_____ 5. D8 is equal to P88.

I Can Do

Cloud Process

Write each step in finding the position / location in the given set of data using the cloud below.
Add or delete clouds, if necessary (10 points).

Solve each item carefully.

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.
a. Find the lower quartile and upper quartile of the data.
b. How many customers are below Q1 ? are above Q3 in terms of each hour in a day?
c. Which hour of the day has less customers? has more customers?

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2. Mrs. Marana is a veterinarian. One morning, she asked her secretary to record the service
time for 15 customers. The following are service times in minutes.
20, 35, 55, 28, 46, 32, 25, 56, 55, 28, 37, 60, 47, 52, 17
Find the value of the 2nd decile, 6th decile, and 8th decile.
3. The scores of Miss World candidates from seven judges were recorded as follows:
8.45, 9.20, 8.56, 9.13, 8.67, 8.85, and 9.17.

a. Find the 60th percentile or P60 of the judges’ scores.

b. What is the P35 of the judges’ scores?

Additional Activities

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
1 4 3
Given: Scores 5, 7, 12, 14, 15, 22, 25, 30, 36, 42, 53, 65

Across
2. D8
65(𝑛+1)
4.
100
90(𝑛+1)
8. 100

9. P9
Down
1. Q2
90(𝑛+1)
3. 100

5. P40
6. P52
7. P54

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ADDITIONAL ACTIVITY ASSESSMENT WHAT I CAN DO
1. a. 𝑄1 = 9; 𝑄3 = 14
b. Below 𝑄1 is 13
customers while
above 𝑄3 is 31
customers
c. 6th hour, 5th hour
2. 𝐷2 = 25;
𝐷6 = 47;
𝐷8 = 55
3. a. 9.13
b. 8.67
WHAT I HAVE WHAT’S NEW WHAT I KNOW
LEARNED Answers May Vary
1. B
1. True WHAT’S MORE 2. A
2. False 1. a. 𝑄1 = 11.75; 𝑄2 = 13, 3. B
3. True 𝑄3 = 14.25 4. D
4. True b. 𝑄1 = 3 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠; 𝑄2 = 6 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠; 5. A
5. False 𝑄3 = 8 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
WHAT’S IN
2. This means that 25% of the learners
finished the test. A low quartile 1. 7
considered good, because it means
the students finished the test in a
2. 8
short period of time. 3. 32
3. a. 𝑃75 = 20 4. 17
b. 4 5. 43
References
Books:
Callanta, Melvin M., et.al. Mathematics 10 Learner’s Module. Rex Book Store Inc., 2015.

Callanta, Melvin M., et.al. Mathematics 10 Teacher’s Guide. Rex Book Store Inc., 2015.

Calpul, Ernest A., et. al. Next Generation Math.

Websites:
https://approved-guide.com/private-schools-can-offer-your-son-or-daughter-having-a-better-
education.html?fbclid=IwAR3JivRV0r9YtfFdC-fGup6fbbl0mEZqwGV4sZ9hc5XzNkXzsqd8XLbFTsg

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