10
Mathematics
Quarter 4 – Module 5:
Using Appropriate Measures of
Position and Other Statistical
Methods in Analyzing and
Interpreting Research Data
Department of Education ● Republic of the Philippines
Mathematics- Grade 10
Alternative Delivery Mode
Quarter 4 – Module 5: Uses Appropriate Measures of Position and Other Statistical
Methods in Analyzing and Interpreting Research Data
First Edition, 2020
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Analyzing and Interpreting
Data Using Quartile
Lesson
1
What’s New
My dear learner, before we start this lesson, let us consider the situation below.
Situation:
In a class of 30 students, the following are their scores during the 3 rd quarter exam.
55
24
25
40
37
36
20
46
41
39
36
41
45
49
52
24
53
29
31
42
36
56
57
39
44
30
44
25
48
46
The teacher wants to give bonus points to the top 25% of the students, and remedial
instruction to the bottom 25% of the students.
a. What are the scores of students who will receive bonus points?
b. What are the scores of students who will undergo remedial instruction?
To answer the questions in the situation above, we have to read and study the discussion
of this lesson in the next part.
What Is It
You learned in the previous modules about the measures of position: quartile, decile,
and percentile.
When will we use quartile in a situation?
Quartile
If the situation or problem involves 25% or 50% or 75% of the data, the best measure
of position to use is the quartile.
In quartiles, the distribution of data is divided into four equal parts. 25% of the
distribution are below the 1st quartile (Q1), 50% are below the 2nd quartile (Q2) and 75% are
below the 3rd quartile (Q3). This means that 25% of the data has a value Q1, 50% of the data
has a value Q2, and 75% of the data has a value Q3.
1
75%
50%
25%
Q1
Q2
Middle 50%
Bottom 25%
Position of quartiles
in the distribution
Q3
Top 25%
Quartiles represent the value for which 25% of the data is below 1 st quartile (Q1) and
the value for which 25% of the data is above 3rd quartile (Q3).
The bottom 25% of the data has a value of
value of > Q3.
Q1, and the top 25% of the data has a
Quartile of Ungrouped Data
To find quartile for ungrouped data, we will solve first its position in the distribution
by using the following formula:
𝒌
Position of 𝑸𝒌 = (𝒏 + 𝟏)
𝟒
where
k is 1, 2, or 3 from Q1, Q2, and Q3
n is the total number of data
Example: Let us answer the questions in the situation given in What’s New.
In a class of 30 students, the following are their scores during the 3 rd quarter exam.
55
40
20
39
45
24
31
56
44
25
24
37
46
36
49
53
42
57
30
48
25
36
41
41
52
29
36
39
44
46
The teacher wants to give bonus points to the top 25% of the students, and remedial
instruction to the bottom 25% of the students.
a. What are the scores of students who will receive bonus points?
b. What are the scores of students who will undergo remedial instruction?
Solution:
Arrange the data into ascending order.
20
24
24
25
25
29
30
31
36
36
36
37
39
39
40
41
41
42
44
44
45
46
46
48
49
52
53
55
56
57
Since the teacher is looking for the top 25% and bottom 25% of students, we will use
quartiles since it involves 25% of the data. This means we are going to find the 1 st quartile (Q1)
and 3rd quartile (Q3).
2
Use the following formula to find the position of k th quartile (Qk)
𝒌
Position of 𝑸𝒌 = (𝒏 + 𝟏)
𝟒
a. The students who will receive bonus points are those students on the top 25% which
are the scores above the value of 3 rd quartile (Q3).
To find the value of the 3rd quartile, let us follow the following steps.
Steps
Step 1
Solve
for the
position
of Q3
Figure/ Expression
Discussion
k
Position of Q k = (n + 1)
4
Position of Q3 =
Substitute:
k with 3 since we are looking for 3rd quartile, and
n with 30 (the total number of students or
number of data)
Perform the addition inside the grouping symbol
( ). (30+1) is 31
3
(30 + 1)
4
3
Position of Q3 = 4 (31)
Position of Q3 =
Use this formula for the kth quartile
93
4
Multiply 3 and 31 which is equal to 93
Divide 93 by 4 resulting to 23.25.
This means, the position of Q3 is between 23rd
and 24th data.
Position of Q3 = 23.25
20
24
24
25
25
29
30
31
36
36
36
37
39
39
40
41
41
42
44
44
45
46
46
48
49
52
53
55
56
57
23rd
24th
Steps
Figure/ Expression
Step 2
Interpolate the value to obtain the 3rd quartile
Discussion
Since the result of the position of
Q3 is a decimal number,
interpolation is needed.
Step 2.1
Subtract 23rd data from
the 24th data.
Step 2.2
Multiply the result in step
2.1 by the decimal part in
the position of Q3.
Step 2.3
Add the result in step 2.2
to the 23rd data.
Look at the arranged data.
The 23rd data is 46 and the 24th
data is 48.
Therefore
48 – 46 = 2
The result in step 2.1 is 2 and the
decimal part in the position of Q3
is 0.25 (from 23.25).
2(0.25) = 0.5
46 + 0.5 = 46.5
3rd quartile (Q3) = 46.5
We used 23rd data since the Q3 is
between 23rd and 24th data, and
the 23rd data is 46.
The value of the 3rd quartile is the
result of step 2.3.
The 3rd quartile here means 75% of the scores are equal or below 46.5. So the scores
above 46.5 are the top 25% of the scores.
Therefore, the students who will receive bonus points are those students got the scores
greater than 46.5.
3
b. The students who will undergo remedial instruction are those students whose scores belong
to the bottom 25% which are the scores equal or below the value of 1 st quartile (Q1).
To find the value of the 1st quartile, let us follow the following steps
Steps
Step 1
Solve for the position
of Q1
Figure/ Expression
k
Position of Q k = (n + 1)
4
1
4
Position of Q k = (30 + 1)
1
4
Position of Q k = (31)
Position of Q k =
31
4
Position of Q k = 7.75
Discussion
Use this formula to find the
position of kth quartile
Substitute
k with 1 since we are looking for
1st quartile, and
n with 30 (the total number of
students or number of data)
Perform the addition inside the
grouping symbol ( ).
(30+1) is 31.
Multiply 1 and 31 which is equal
to 31.
Divide 31 by 4 resulting to 7.75.
This means, the position of Q1 is
between 7th and 8th data.
7th
8th
20
24
24
25
25
29
30
31
36
36
36
37
39
39
40
41
41
42
44
44
45
46
46
48
49
52
53
55
56
57
Step 2
Interpolate the value to obtain the 1 st quartile
Step 2.1
Subtract 7th data from the 8th
data.
Step 2.2
Multiply the result in step 2.1
by the decimal part in the
position of Q1.
Step 2.3
Add the result in step 2.2 to
the 7th data.
Therefore
31 – 30 = 1
1(0.75) = 0.75
30 + 0.75 = 30.75
1st quartile (Q1) =
30.75
Since the result of the position of
Q1 is a decimal number,
interpolation is needed.
Look at the arrange data.
The 7th data 30 and
the 8th data is 31.
The result in step 2.1 is 1 and the
decimal part in the position of Q1
is 0.75 (from 7.75).
We used 7th data since the Q1 is
between 7th and 8th data, and the
7th data is 30.
The value of the 1st quartile is the
result of step 2.3.
Since the value of the 1st quartile is 30.75, it means 25% of the scores of students
are equal or below 30.75.
Therefore, the students who will undergo remedial instruction are those students whose
scores are equal or below 30.75 .
4
Quartile of Grouped Data
To find the quartile of a grouped data, we will be using the following formulae:
•
First, we have to solve for the class of the quartile using this formula
𝐐𝐤 𝐜𝐥𝐚𝐬𝐬 =
Where:
•
𝐤𝐍
𝟒
k is nth quartile, where n = 1, 2, and 3
N is the total frequency
Then, use the following formula to find the value of the k th quartile (Qk)
𝐤𝐍
− 𝐜𝐟𝐛
𝐐𝐤 = 𝐋𝐁 + ( 𝟒
)𝒊
𝐟𝐐𝐤
Where:
LB = lower boundary of the Qk class
N = total frequency
cfb = cumulative frequency of the
class before the Qk class
= frequency of the Qk class
= size of class interval
= nth quartile, where
n = 1, 2, and 3
k
Example:
The table shows the scores of 50 students in a 60-item Mathematics long exam.
Scores
56-60
51-55
46-50
41-45
36-40
31-35
26-30
Frequency
4
9
10
9
8
7
3
The passing score of this exam is 45. Is half or 50% of the class passed the exam?
Solution:
Let us put another columns to our table for the lower boundaries (LB) and the
cumulative frequency (cf).
Class Interval
Scores
Frequency
(f)
56-60
51-55
46-50
41-45
36-40
31-35
26-30
4
9
10
9
8
7
3
N = 50
Lower
Boundaries
Less than
Cumulative Frequency
(LB)
(<cf)
55.5
50.5
45.5
40.5
35.5
30.5
25.5
50
46
37
27
18
10
3
5
•
How does the Lower Boundaries (LB) computed?
The lower boundary of an interval is computed by subtracting 0.5 to the smallest
number in the interval.
In interval 56-60, the smallest number is 56. Let us use 56 then subtract it by 0.5.
So, 56 – 0.5 = 55.5
In interval 51-55, the smallest number is 51. Subtract 51 by 0.5 resulted to 50.5.
Class Interval
Scores
Lower Boundaries
(LB)
56-60
51-55
46-50
41-45
36-40
31-35
26-30
55.5
50.5
45.5
40.5
35.5
30.5
25.5
56 – 0.5 = 55.5
51 – 0.5 = 50.5
46 – 0.5 = 45.5
41 – 0.5 = 40.5
36 – 0.5 = 35.5
31 – 0.5 = 30.5
26 – 0.5 = 25.5
a. How does the Cumulative Frequency (cf) computed?
The cumulative frequency is the total frequency from the bottom to the current
interval. To compute, start from the bottom interval.
The cumulative frequency of the bottom interval is equal to its frequency. In the
table below, the frequency of the bottom class interval (26-30) is 3, then its cf is also 3.
For the next interval (31-35), add the cf of the interval 26-30 to the frequency of
the interval 31-35. The frequency of the interval 31-35 is 7. So the cf of this interval is
7 + 3 = 10.
For the next intervals, just follow the step where add the frequency of the current
interval by the cumulative frequency of the previous interval.
Class
Interval
Scores
56-60
51-55
46-50
41-45
36-40
31-35
26-30
Frequency
(f)
4
9
10
9
8
7
3
N = 50
f
+
4
9
10
9
8
7
cf
before
the
interval
+
46
+
37
+
27
+
18
+
10
+
3
Copy the frequency
Less than
Cumulative
Frequency
(<cf)
=
=
=
=
=
=
50
46
37
27
18
10
3
Take note that the cumulative frequency of the top interval is equal to the total
frequency (N).
6
If you are done with the table, then let us answer the question: Is half or 50% of the
class passed the exam?
50% of the class are the scores equal or below the value of 2 nd quartile.
So, to solve for this problem, we are going to compute the 2nd quartile.
Steps
Step 1
Solve for the Q2
class
Figure/ Expression
Discussion
Use this formula to find the kth
quartile class where N is the total
frequency.
Substitute k with 2 since we are
looking for 2nd quartile and N with 50.
kN
4
2N 2(50)
Q 2 class =
=
4
4
100
Q 2 class =
4
Q2 class = 25
Q k class =
Multiply 2 and 50 resulting 100.
Divide 100 by 4 equals 25.
Thus the 2nd quartile (Q2) is the 25th score.
This means we need to find the class interval where the 25 th score is contained. Look
at the column of cumulative frequency in the table, the 25 th score is within the 19th and 27th
scores.
Class
Interval
Scores
Frequency
(f)
Lower
Boundaries
(LB)
56-60
51-55
46-50
41-45
36-40
31-35
26-30
4
9
10
9
8
7
3
55.5
50.5
45.5
40.5
35.5
30.5
25.5
Less than
Cumulative
Frequency
(<cf)
50
46
37
27
18
10
3
(19th-27th score)
(11th-18th score)
(4th- 10th score)
(1st -3rd score)
Q2 Class
The 19th -27th scores belong to the class interval 41-45. So, the 25th score is within the
class interval 41-45.
The Q2 class is the class interval 41-45.
After finding the 2nd quartile (Q2) class, we will now solve for the value of the 2 nd
quartile (Q2).
Steps
Step 2
Use the
formula
to find
the value
of Q2
Figure/ Expression
2N
− cfb
Q 2 = LB + ( 4
)𝑖
fQ2
25 − 18
Q 2 = 40.5 + (
)5
9
Discussion
Where:
LB = 40.50 (lower boundary of Q2 class)
𝑐𝑓𝑏 = 18
(cumulative frequency of the class
interval before Q2 class.)
𝑓𝑄 2 = 9 (frequency of Q2 class)
𝑖=5
(the size of class intervals)
Substitute the values above in the formula.
7
7
Q 2 = 40.5 + ( ) 5
9
25 – 18 is 7.
Q 2 = 40.5 + 3.89
7𝑥5 ÷ 9 = 3.88888 rounded to 3.89.
Q 2 = 44.39
Add 40.5 and 3.89 result to 44.39.
Since the value of 𝑄2 = 44.39, this means 50% of the students have a score less than
or equal to 44.39.
Therefore, the 50% of the class did not pass the Mathematics long exam since the
passing score is 45.
What’s More
Let’s practice.
1. The following is the profit of canteen for each food displayed.
Foods
Sandwich
Hotdog
Toron
Banana Cue
Bread
Profit
510
505
580
550
510
Foods
Puto
Cupcake
Pizza
Siomai
Camote Cue
Profit
530
505
500
560
535
If the canteen owner wants to reduce 25% of his food display, which of the
foods is he going to remove?
Answer the following questions.
a. What kind of data is this? Grouped or ungrouped?
b. What measure of position is appropriate to use in this problem? Quartile, decile, or
percentile?
c. If the owner want to reduce his food display by 25%, which foods will he remove: the foods
at the top 25% of the profit or the foods at the bottom 25% of the profit? Why?
d. Arrange the foods according to its profit.
If you will use table in arranging the foods, put the food with the highest profit at the top
and the food with the lowest profit at the bottom.
Foods
Profit
(food with the highest profit)
(food with the lowest profit)
e. Solve for the position of 1st quartile (Q1).
f. Find the value of the 1st quartile (Q1).
g. What are the foods that the canteen owner should remove?
8
2. A barangay captain will sponsor a scholarship program for poor but deserving college
students in their barangay. There are 45 applicants for the said scholarship. The
selection for the scholar will be based on the family’s monthly income. The captain
decided to grant the scholarship to the 25% of the applicant with the lowest family
income.
Monthly Income (in Pesos)
9,001 – 10,000
8,001 – 9,000
7,001 – 8,000
6,001 – 7,000
5,001 – 6,000
Frequency
7
13
15
7
3
Answer the following questions.
a. What kind of data is this? Grouped or ungrouped?
b. What measure of position is appropriate to use in this problem? Quartile, decile, or
percentile?
c. What income would the captain choose, the highest 25% family income of applicants
or the lowest 25%? Why?
d. Complete the table with lower boundaries and cumulative frequency.
e. Solve for the class of 1st quartile (Q1).
f. Find the value of the 1st quartile (Q1).
g. What are the family income of the student applicants that the barangay captain choose
to grant the scholarship?
Now check your answer from the answer key at the back. If you got it correctly, then
you can proceed to the next lesson.
9
Lesson
2
Analyzing and Interpreting
Data Using Decile
What’s New
My dear learner, before we proceed to our discussion of this lesson, let us explore the
following situation.
Situation:
The table shows the scores of 55 students in a 60-item Mathematics long exam.
Scores
56-60
51-55
46-50
41-45
36-40
31-35
26-30
Frequency
5
11
8
7
10
5
9
The passing score of this exam is 45.
a. Are all in the bottom 40% of the class failed in the exam?
b. Are all in the top 30% of the class passed in the exam?
To answer these questions, let us study the discussions that follows.
What Is It
Decile
The deciles are nine score-points which divide a distribution into ten groups. Deciles
are denoted as D1, D2, D3,…, D9. They are computed in the same way as the quartiles are
calculated.
In the figure below, it shows that 10% of the data is less than or equal to the value of
D1, 20% of the data is less than or equal to the value of D2, 30% of the data is less than or equal
to the value of D3,…, 90% of the data is less than or equal to the value of D 9.
When will we use decile in analysing and interpreting data?
10
If the situation or problem is asking for the 10%, 20%, 30%, 40%, 50%, 60%, 70%,
80%, and 90% of the data, then the appropriate measure of position to be use is decile.
If the situation or problem is asking for the top 10% of the data, it means we are looking
for the data above the 90% of data or the value of 9th decile (D9). If it is asking for the top 20%
of the data, it means we are looking for the data above the 80% of the data or the value of 8 th
decile (D8). It follows to other top percentage of data.
Decile for Ungrouped Data
To find the value of decile, we have to solve first its position in the distribution.
Let us use the following formula to solve for the position of decile for ungrouped data.
Position of 𝐃𝐤 =
𝐤
𝟏𝟎
( 𝐧 + 𝟏)
where k is 1, 2, 3, …, 9 from D1, D2, …, and D9
n is the total number of data
Let’s take this situation.
Example:
In a class of 40 students, the following are their scores during the 4 th quarter exam.
55
24
25
22
40
37
36
35
20
46
41
39
39
36
41
28
45
49
52
35
24
53
29
42
31
42
36
23
56
57
39
37
44
30
44
37
25
48
46
40
The teacher wants to give bonus points to the top 10% of the students, and remedial
instruction to the bottom 20% of the students.
Step
Step 2
Find
the
value
of D9
Figure/Expression
20
22
23
24
24
25
25
28
29
30
31
35
35
36
36
36
37
37
37
39
39
39
40
40
41
41
42
42
44
44
45
46
46
48
49
52
53
55
56
57
37th
Discussion
Using the position of
9th decile (D9) which
is 37, count the
arranged data in
ascending order up to
the 37th.
The 37th score is 53.
So, the value of 9th
decile (D9) is 53.
a. What are the scores of students who will receive bonus points?
b. What are the scores of students who will undergo remedial instruction?
11
Solution:
Arrange the data into ascending order.
20
22
23
24
24
31
35
35
36
36
39
39
40
40
41
45
46
46
48
49
25
36
41
52
25
37
42
53
28
37
42
55
29
37
44
56
30
39
44
57
Since the teacher is looking for the top 10% and bottom 20% of students, this means
we are going to find the 9th decile (D9) and 2nd decile (D2).
The 90% of data is less than or equal to the value of 9th decile, and the other 10% of
the data (which is the top 10%) are greater than the value of the 9 th decile. Therefore, we will
solve for the 9th decile (D9) to find the top 10% of the students’ scores.
The 20% of the data is less than or equal to the value of 2nd decile. Therefore, we will
solve for the 2nd decile (D2) to find the bottom 20% of the students’ scores.
To find the 9th decile (D9) and 2nd decile (D2), we have to solve first its position in the
data.
Use the following formula to find the position of k th decile (Dk)
Position of Dk =
k
10
(n + 1)
Let us now answer the questions.
a. The students who can received a bonus points from the teacher are the students with
scores above 90% or above 9th decile (D9).
To find the 9th decile, let us find first its position:
Step
Figure/Expression
Step 1
k
Solve for
Position of Dk = (n + 1)
10
the position
of D9
Position of D9 =
9
(40
10
Position of D9 =
9
(41)
10
Position of D9 =
369
10
+ 1)
Position of D9 = 36.9 or 37
Now let us find the value of the 9th decile.
12
Discussion
Use this formula where
k is 9 since we are looking for the
9th decile, and
n is 40 (the total number of
students).
Substitute the values of k and n
stated above.
Perform the addition inside the
grouping symbol ( ).
(40+1) is 41.
Multiply 9 and 41 which is equal to
369.
Divide 369 by 10 resulting to 36.9,
rounded to 37.
Step
Step 2
Find the
value of
D9
Figure/Expression
20 22 23 24 24 25 25 28 29 30
31 35 35 36 36 36 37 37 37 39
39 39 40 40 41 41 42 42 44 44
45 46 46 48 49 52 53 55 56 57
Discussion
Using the position of 9th
decile (D9) which is 37,
count the arranged data in
ascending order up to the
37th.
The 37th score is 53.
So, the value of 9th decile
(D9) is 53.
37th
Therefore, the students who can received the bonus points are those with scores more
than 53.
b. The students who will undergo remedial instruction are those with scores below the
2nd decile (D2).
To find the value of 2nd decile, let us first solve its position in the arrange data.
Step
Step 1
Solve for
the position
of D2
Figure/Expression
Position of Dk =
Position of D2 =
Position of D2 =
Position of D2 =
k
10
2
10
(n + 1)
(40 + 1)
2
10
Discussion
Use this formula where
k is 2 since we are looking for the
9th decile, and
n is 40 (the total number of
students).
Substitute the values of k and n
stated above.
Perform the addition inside the
grouping symbol ( ).
(40+1) is 41.
Multiply 2 and 41 which is equal to
82.
Divide 82 by 10 resulting to 8.2,
rounded to 8.
(41)
82
10
Position of 𝐷2 = 8.2 or 8
Now let us find the value of the 2nd decile (D2).
Step
Figure/Expression
Step 2
8th
Find the
value of
D2
20 22 23 24 24 25 25 28 29 30
Discussion
Using the position of 2nd
decile (D2) which is 8,
count the arranged data in
ascending order up to the
8th.
31 35 35 36 36 36 37 37 37 39
The 8th score is 28.
39
39
40
40
41
41
42
42
44
44
45
46
46
48
49
52
53
55
56
57 (D2) is 28.
So, the value of 2nd decile
Therefore, the students who will undergo remedial instruction are those with scores
less than or equal to 28.
13
Decile for Grouped Data
To find the decile of a grouped data, we will use the following formulae:
•
First, we have to solve for the class of the decile using this formula.
𝐃𝐤 𝐜𝐥𝐚𝐬𝐬 =
Where:
•
𝐤𝐍
𝟏𝟎
k is the nth decile, where n = 1, 2, 3,…, and 9
N is the total frequency
Then, use the following formula to find the value of the k th decile (Dk).
𝐤𝐍
− 𝐜𝐟𝐛
𝐃𝐤 = 𝐋𝐁 + ( 𝟏𝟎
)𝒊
𝐟𝐃𝐤
Where:
LB = lower boundary of the Dk class
N = total frequency
cfb = cumulative frequency of the
class before the Dk class
fDk = frequency of the Dk class
𝑖
= size of class interval
k
= nth decile, where n = 1,
2, 3,…, and 9
Example: Let us answer the situation given in What’s New.
The table shows the scores of 55 students in a 60-item Mathematics long exam.
Scores
56-60
51-55
46-50
41-45
36-40
31-35
26-30
Frequency
5
11
8
7
10
5
9
The passing score of this exam is 45.
a. Are all in the bottom 40% of the class failed in the exam?
b. Are all in the top 30% of the class passed in the exam?
Solution:
Let us add another columns to our table for the lower boundaries and the cumulative
frequency.
Class Interval
Scores
Frequency
(f)
56-60
51-55
46-50
41-45
36-40
31-35
26-30
5
11
8
7
10
5
9
N = 55
Lower
Boundaries
(LB)
55.5
50.5
45.5
40.5
35.5
30.5
25.5
14
Less than
Cumulative Frequency
(<cf)
55
50
39
31
24
14
9
Let us now answer the questions.
a. The bottom 40% of the class are those students with scores less than or equal to
the value of 4th decile (D4).
Let’s find 4th decile (D4) with the following steps.
Steps
Step 1
Solve for
the 4th decile
(D4) class
Figure/Expression
Dk class =
kN
10
4(55)
10
220
D4 class =
10
D4 class = 22
D4 class =
Discussion
Use this formula where
k is 4 since we are looking for D4, and
N is 55 (total frequency).
Substitute k with 4 and N with 55.
Multiply 4 and 55 resulting to 220.
Divide 220 by 10 is equal to 22.
This 4th decile class means that the value of 4th decile is within the interval where 22nd
data belong.
Let us now look for the class interval where 22nd data belong.
D4
class
Class
Interval
Scores
Frequency
(f)
Lower
Boundaries
(LB)
56-60
5
55.5
Less than
Cumulative
Frequency
(<cf)
55
51-55
11
50.5
50
46-50
8
45.5
39
41-45
7
40.5
31
36-40
10
35.5
24
31-35
5
30.5
14
15th to 24th
scores
th
10 to 14th
26-30
9
25.5
9
1st to 9th
22nd score
belonged
N = 55
As you can see above, the 22nd data is within 15th to 24th data and 15th to 24th data
belonged to the interval 36-40.
We will use this interval as our reference for the formula to find the value of 4 th decile.
In the formula, we will use the lower boundary (LB), frequency (f Dk) of the interval and the
cumulative frequency before the class interval (cfb). Look again the table above, the encircled
numbers are the values of LB, fDk, and cfb.
Let us now solve for the value of 4th decile (D4) using the formula
kN
−cfb
Dk = LB + ( 10f
Dk
)𝑖
15
where k is 4.
Step
Step 2
Use the
formula
to find
the value
of D4
Figure/Expression
Discussion
Where:
LB = 35.5 (lower boundary of D4 class)
4N
4N
10
− cfb
10
cf
b
D4 = LB + (
)𝑖
fD4
fD4
D4 = 35.5 + (
22−14
)5
10
8
10
40
10
= 22 (the D4 class from step 1)
= 14 (cumulative frequency before the class
interval of D4)
= 10 (Frequency of the class interval)
= 5 (Class interval size)
Substitute the values above in the formula.
D4 = 35.5 + ( ) 5
(22 – 14) is 8.
D4 = 35.5 +
8 x 5 = 40
D4 = 35.5 + 4
40 ÷ 10 = 4
D4 = 39.5
35.5 + 4 = 39.5
The value of 4th decile is 39.5. This means that the bottom 40% scores are below or
equal to 39.5.
Therefore, all the students with scores in the bottom 40% are all failed in the exam
since the passing score is 45.
b. The top 30% of the class are the scores above the 70% or the 7th decile (D7).
Let’s find 7th decile (D7) with the following steps.
Steps
Step 1
Solve for
the 7th decile
(D7) class
Discussion
Figure/Expression
Dk class =
D7 class =
D7 class =
𝑘𝑁
10
7(55)
10
385
10
D7 class = 38.5
Use this formula where
k is 7 since we are looking for D7, and
N is 55 (total frequency).
Substitute k with 7 and N with 55.
Multiply 7 and 55 resulting to 385.
Divide 385 by 10 is equal to 38.5.
This 7th decile class means that the value of 7th decile is within the interval where 38.5th
data belong.
Let us now look for the class interval where 38.5th data belong.
16
D7
class
Class
Interval
Scores
Frequency
(f)
Lower
Boundaries
(LB)
56-60
5
55.5
Less than
Cumulative
Frequency
(<cf)
55
51-55
11
50.5
50
46-50
8
45.5
39
41-45
7
40.5
31
36-40
10
35.5
24
31-35
5
30.5
14
25.5
9
26-30
9
N = 55
32nd to 39th
scores
38.5th
score
belonged
As you can see in the table, the 38.5th data is within 32nd to 39th data and 32nd to 39th
data belonged to the interval 46-50.
We will use this interval as our reference for the formula to find the value of 7 th decile.
In the formula, we will use the lower boundary (LB), frequency (f Dk) of the interval and the
cumulative frequency before the class interval (cfb). Look again the table above, the encircled
numbers are the values of LB, fDk, and cfb.
Let us now solve for the value of 7th decile (D7) using the formula
𝑘𝑁
Dk = LB + ( 10
−𝑐𝑓𝑏
𝑓𝐷𝑘
Step
)𝑖
where k is 7.
Discussion
Figure/Expression
Step 2
Use the
formula
to find
the value
of D7
Where:
LB = 45.5 (lower boundary of D7 class)
7𝑁
10
= 38.5 (the D7 class from step 1)
𝑐𝑓𝑏 = 31 (cumulative frequency before the class
interval of D7)
𝑓𝐷4 = 8 (Frequency of the class interval)
𝑖 = 5 (Class interval size)
7𝑁
D7 = LB + ( 10
−𝑐𝑓𝑏
𝑓𝐷7
D7 = 45.5 + (
)𝑖
38.5−31
)5
8
7.5
8
37.5
8
Substitute the values above in the formula.
38.5 – 31 = 7.5
D7 = 45.5 + ( ) 5
7.5 x 5 = 37.5
D7 = 45.5 +
37.5 ÷ 8 = 4.69
D7 = 45.5 + 4.69
45.5 + 4.69 = 50.19
The value of 7th decile is 50.19. This means that the scores above 7 th decile or the top
30% are greater than 50.19.
Therefore, all students whose scores belong to the top 30% passed the exam.
17
What’s More
Let’s practice.
1. The following is the profit of canteen for each food displayed.
Foods
Sandwich
Hotdog
Toron
Banana Cue
Bread
Profit
510
505
580
550
510
Foods
Puto
Cupcake
Pizza
Siomai
Camote Cue
Profit
530
505
500
560
535
If the canteen owner wants to increase the capital of the 40% of his food
display, which of the foods will he choose?
Answer the following questions.
a. What kind of data is this? Grouped or ungrouped?
b. What measure of position is appropriate to use in this problem?
Quartile, decile, or percentile?
c. If the owner wants to increase the capital of the 40% of his food display, which foods will
he choose: the foods at the top 40% of the profit or the foods at the bottom 40% of the profit?
Why?
d. Arrange the foods according to its profit.
e. Solve for the position of 6th decile (D6).
f. Find the value of the 6th decile (D6).
g. What are the foods that the canteen owner should increase the capital?
2. A barangay captain will sponsor a scholarship program for poor but deserving college
students in their barangay. There are 50 applicants for the said scholarship. The
selection for the scholars will be based on their family’s monthly income. The captain
decided to grant the scholarship to the 20% of the applicant with the lowest family
income.
Monthly Income (in Pesos)
9,001 – 10,000
8,001 – 9,000
7,001 – 8,000
6,001 – 7,000
5,001 – 6,000
Frequency
8
13
15
9
5
Answer the following questions.
a. What kind of data is this? Grouped or ungrouped?
b. What measure of position is appropriate to use in this problem? Quartile, decile, or
percentile?
c. What income would the captain choose, the highest 20% family income of applicants
or the lowest 20%? Why?
d. Complete the table with lower boundaries and cumulative frequency.
e. Solve for the class of 2nd decile (D2).
f. Find the value of the 2nd decile (D2).
g. What are the family income of the student applicants that the barangay captain choose
to grant the scholarship?
18
Lesson
Analyzing and Interpreting
Data Using Percentile
3
What’s New
My dear learner, before our discussion of this lesson, let us explore the
following activities.
Situation:
In a class of 40 students, the following are their scores during the 1 st quarter exam.
55
24
25
22
40
37
36
35
20
46
41
39
39
36
41
28
45
49
52
35
24
53
29
42
31
42
36
23
56
57
39
37
44
30
44
37
25
48
46
40
The teacher wants to give bonus points to the top 15% of the students, and remedial
instruction to the bottom 35% of the students.
a. What are the scores of students who will receive bonus points?
b. What are the scores of students who will undergo remedial instruction?
What Is It
Percentile
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. The 1 st percentile (P1) separates
the lowest 1% from the other 99%, the 2nd percentile (P2) separates the lowest 2% from the
other 98%, and so on.
P25
Q1
P10
D1
P20
D2
P50
Q2
P30
D3
P40
D4
P50
D5
P75
Q3
P60
D6
P70
D7
P80
D8
P90
D9
The percentiles determine the value for 1%, 2%,…, and 99% of the data. P 20 or 20th
percentile of the data means 20% of the data have values less than or equal to P20.
The 1st quartile is the 25th percentile. It means 25% of the data is less than or equal to
the value of P25 or Q1.
The 1st decile is the 10th percentile. It means 10% of the data is less than or equal to the
value of P10 or D1, and so on.
19
Percentile for Ungrouped Data
Here is how to find percentile for ungrouped data. We will solve first its position by
using the following formula:
Position of 𝐏𝐤 =
𝐤
𝟏𝟎𝟎
(𝐧 + 𝟏)
where k is 1, 2, 3, …, 99 from P1, P2, …, and P99
n is the total number of data
Example:
In a class of 40 students, the following are their scores during the 1 st quarter exam.
55
24
25
22
40
37
36
35
20
46
41
39
39
36
41
28
45
49
52
35
24
53
29
42
31
42
36
23
56
57
39
37
44
30
44
37
25
48
46
40
The teacher wants to give bonus points to the top 15% of the students, and remedial
instruction to the bottom 35% of the students.
a. What are the scores of students who will receive bonus points?
b. What are the scores of students who will undergo remedial instruction?
Solution:
Arrange the data into ascending order.
20
22
23
24
24
31
35
35
36
36
39
39
40
40
41
45
46
46
48
49
25
36
41
52
25
37
42
53
28
37
42
55
29
37
44
56
30
39
44
57
Since the teacher is looking for the top 15% and bottom 35% of students, this means
we are going to find the 85th percentile (P85) and 35th percentile (P35).
The 85% of data is less than or equal to the value of 85 th percentile, and the other 15%
of the data (which is the top 15%) are greater than the value of the 85 th percentile. Therefore,
we will solve for the 85th percentile (P85) to find the top 15% of the students’ scores.
The 35% of the data is less than or equal to the value of 35th percentile. Therefore, we
will solve for the 35th percentile (P35) to find the bottom 35% of the students’ scores.
To find the 85th percentile (P85) and 35th percentile (P35), we have to solve first its
position in the data.
Use the following formula to find the position of k th percentile (Pk)
Position of Pk =
k
100
(n + 1)
Let us now answer the questions.
a. The students who can received a bonus points from the teacher are the students with
scores above 85% or 85th percentile (P85).
20
To find the 85th percentile (P85), let us first find its position:
Step
Figure/Expression
Step 1
Solve for
k
Position of Pk =
(n + 1)
the
100
position of
P85
Position of P85 =
85
(40
100
Position of P85 =
85
(41)
100
Position of P85 =
3485
100
Discussion
Use this formula where
k is 85 since we are looking for the
85th percentile, and
n is 40 (the total number of
students).
Substitute the values of k and n
stated above.
Perform the addition inside the
grouping symbol ( ).
(40+1) is 41.
Multiply 85 and 41 which is equal
to 3485.
Divide 3485 by 100 resulting to
34.85, rounded to 35.
+ 1)
Position of P85 = 34.85 or 35
Now let us find the value of the 85th percentile.
Step
Step 2
Find the
value of
P85
Figure/Expression
20
22
23
24
24
25
25
28
29
30
31
35
35
36
36
36
37
37
37
39
39
39
40
40
41
41
42
42
44
44
45
46
46
48
49
52
53
55
56
57
35th
Discussion
Using the position of
85th percentile (P85)
which is 35, count the
arranged data in
ascending order up to the
35th.
The 35th score is 49.
So, the value of 85th
percentile (P85) is 49.
Therefore, the students who can received the bonus points are those with scores
greater than 49. The scores are 52,53,55,56 and 57.
b. The students who will undergo remedial instruction are those with scores below the
35th percentile (P35).
To find the value of 35th percentile (P35), let us first solve its position in the arrange
data.
Step
Step 1
Solve
for the
position
of P35
Figure/Expression
Position of 𝑃𝑘 =
𝑘
(𝑛
100
Position of 𝑃35 =
Position of 𝑃35 =
Position of 𝑃35 =
35
100
+ 1)
(40 + 1)
35
(41)
100
1435
100
Position of 𝑃35 = 14.35 or 14
Discussion
Use this formula where
k is 35 since we are looking for the 35th
percentile, and
n is 40 (the total number of students).
Substitute the values of k and n stated
above.
Perform the addition inside the grouping
symbol ( ).
(40+1) is 41.
Multiply 35 and 41 which is equal to 1435.
Divide 1435 by 100 resulting to 14.35,
rounded to 14.
21
Now let us find the value of the 35th percentile (P35).
Step
Step 2
Find the
value of
P35
Figure/Expression
20
22
23
24
24
25
25
28
29
30
31
35
35
36
36
36
37
37
37
39
Discussion
Using the position of 35th
percentile (P35) which is
14, count the arranged data
in ascending order up to
the 14th.
39
39
40
40
41
41
42
42
44
44
The 14th score is 36.
45
46
46
48
49
52
53
55
56
57
So, the value of 35th
percentile (P35) is 36.
14th
Therefore, the students who will undergo remedial instruction are those with scores
less than or equal to 36 that is, students with scores of 20,22,23,24,25,28,29,30,31,35,and 36.
Percentile for Grouped Data
To find the percentile of a grouped data, we will use the following formulae:
•
First, we have to solve for the class of the percentile using this formula.
Pk class =
Where:
•
𝐤𝐍
𝟏𝟎𝟎
k is the nth percentile, where n = 1, 2, 3,…, and 99
N is the total frequency
Then, use the following formula to find the value of the kth percentile (Pk).
𝐤𝐍
−𝐜𝐟𝐛
𝟏𝟎𝟎
Pk = LB + (
Where:
LB = lower boundary of the Pk class
N = total frequency
cfb = cumulative frequency of the
class before the Pk class
𝐟𝐏𝐤
)𝐢
𝑓𝑃 𝑘 = frequency of the Pk class
𝑖
= size of class interval
𝑘
= nth percentile, where n = 1, 2, 3,…,
and 99
Example:
The table shows the scores of 55 students in a 60-item Mathematics long exam.
Scores
56-60
51-55
46-50
41-45
36-40
31-35
26-30
Frequency
5
11
8
7
10
5
9
The passing score of this exam is 45.
22
a. Are all in the bottom 45% of the class failed in the exam?
Solution:
Let us put another columns to our table for the lower boundaries and the cumulative
frequency.
Class
Interval
Scores
Frequency
(f)
Lower
Boundaries
(LB)
Less than
Cumulative
Frequency
(<cf)
56-60
51-55
46-50
41-45
36-40
31-35
26-30
5
11
8
7
10
5
9
55.5
50.5
45.5
40.5
35.5
30.5
25.5
55
50
39
31
24
14
9
N = 55
Let us now answer the questions.
a. The bottom 45% of the class are those students with scores less than or equal to
the value of 45th percentile (P45).
Let’s find 45th percentile (P45) with the following steps.
Steps
Step 1
Solve for
the 45th
percentile
(P45) class
Discussion
Figure/Expression
Pk class =
kN
100
P45 class =
P45 class =
45(55)
100
2475
100
P45 class = 24.75
Use this formula where
k is 45 since we are looking for P45, and
N is 55 (total frequency).
Substitute k with 45 and N with 55.
Multiply 45 and 55 resulting to 2475.
Divide 2475 by 100 is equal to 24.75.
This 45th percentile class means that the value of 45th percentile is within the interval
where 24.75th data belong.
Let us now look for the class interval where 24.75 th data belong. Since 24.75th is in the
boundary between 24th and 25th, then let us round off 24.75 to 25. Now, 45 th percentile belongs
to the interval 41-45.
23
P45
class
Class
Interval
Scores
Frequency
(f)
Lower
Boundaries
(LB)
56-60
5
55.5
Less than
Cumulative
Frequency
(<cf)
55
51-55
11
50.5
50
46-50
8
39
41-45
7
45.5
40.5
36-40
31-35
10
5
35.5
30.5
24
14
25th to 31st
scores
15th to 24th
10th to 14th
26-30
9
N = 55
25.5
9
1st to 9th
31
25th score
belonged
We will use this interval as our reference for the formula to find the value of 45 th
percentile. In the formula, we will use the lower boundary (LB), frequency (f Pk) of the interval
and the cumulative frequency before the class interval (cf b). Look again the table above, the
encircled numbers are the values of LB, fPk, and cfb.
Let us now solve for the value of 45th percentile (P45) using the formula
kN
−cfb
Pk = LB + (100
fPk
Step
Step 2
Use the
formula
to find
the value
of P45
)i
where k is 45.
Discussion
Figure/Expression
45N
P45 = LB + ( 100
−cfb
fP45
)i
Where:
LB = 40.5 (lower boundary of D4 class)
45𝑁
100
= 24.75 (the P45 class from step 1)
𝑐𝑓𝑏 = 24 (cumulative frequency before the
class interval of P45)
𝑓𝑃45 = 7 (Frequency of the class interval)
𝑖 = 5 (Class interval size)
P45 = 40.5 +
24.75−24
(
)5
7
P45 = 40.5 + (
P45 = 40.5 +
0.75
)5
7
3.75
7
Substitute the values above in the formula.
(24.75 – 24) is 0.75.
0.75 x 5 = 3.75
P45 = 40.5 + 0.54
3.75 ÷ 7 = 0.54
P45 = 41.04
40.5 + 0.54 = 41.04
The value of 45th percentile is 41.04. This means that the bottom 45% scores are below
or equal to 41.04.
Therefore, all the students with scores in the bottom 45% are all failed in the exam
since the passing score is 45.
24
What’s More
Let’s practice.
1. The following is the profit of canteen for each food displayed.
Foods
Sandwich
Hotdog
Toron
Banana Cue
Bread
Profit
510
505
580
550
510
Foods
Puto
Cupcake
Pizza
Siomai
Camote Cue
Profit
530
505
500
560
535
If the canteen owner wants to reduce 15% of his food display, which of the
foods is he going to remove?
Answer the following questions.
a. What kind of data is this? Grouped or ungrouped?
b. What measure of position is appropriate to use in this problem? Quartile, decile, or
percentile?
c. If the owner want to reduce his food display by 15%, which foods will he remove: the foods
at the top 15% of the profit or the foods at the bottom 15% of the profit? Why?
d. Arrange the foods according to its profit.
e. Solve for the position of 15th percentile (P15).
f. Find the value of the 15th percentile (P15).
g. What are the foods that the canteen owner should remove?
2. A barangay captain will sponsor a scholarship program for poor but deserving college
students in their barangay. There are 50 applicants for the said scholarship. The
selection for the scholar will be based on the family’s monthly income and their grades.
The captain decided to reduce his choices by removing 55% of the applicants based on
their family income. What family income of the student applicants would the captain
remove from his choices of scholarship?
Monthly Income (in Pesos)
9,001 – 10,000
8,001 – 9,000
7,001 – 8,000
6,001 – 7,000
5,001 – 6,000
Frequency
8
13
15
9
5
Answer the following questions.
a. What kind of data is this? Grouped or ungrouped?
b. What measure of position is appropriate to use in this problem? Quartile, decile, or
percentile?
c. What income would the captain choose, the highest 55% family income of applicants
or the lowest 55%? Why?
d. Complete the table with lower boundaries and cumulative frequency.
e. Solve for the class of 45th percentile (P45).
f. Find the value of the 45th percentile (P45).
g. What are the family income of the student applicants that the barangay captain choose
to remove from his choices of scholarship?
25
What I Have Learned
Let’s see what you have learned by completing the statements below providing the
missing word(s), phrase, or figures.
In quartile, the distribution of data is divided into (1.)_______ equal parts. 25% of the
data has a value (2.)_______, (3.)_______ of the data has a value ≤ Q2, and 75% of the data has
a value ≤ Q3. The value at the bottom 25% of the data is below (4.)_______ and the value at the
top (5.)_______ of the data is above the 3rd quartile.
In decile, the distribution of data is divided into 10 equal parts. 10% of the data has a
value ≤ D1, (6.)_______ of the data has a value ≤ D2, 30% of the data has a value (7.)_______,
…, and (8.)_______ of the data has a value ≤ D 9. The value at the bottom 20% of the data is
below (9.)_______, and the value at the top 10% of the data is (10.)_______ the 9th decile.
In percentile, the distribution of data is divided into 100 equal parts. 10% of the data
has a value ≤ P10, (11.)_______ of the data has a value ≤ P 37, 68% of the data has a value
(12.)_______, …, and (13.)_______ of the data has a value ≤ P 99. The value at the bottom 20%
of the data is below (14.)_______ percentile, and the value at the top (15.)_______ of the data is
above the 85th percentile.
What I Can Do
Collect data about the height of your 15 friends. Present it using a frequency table.
Identify the height of the 10% tallest among your friends, then list down their names.
Assessment
Choose the letter of the best answer.
1. Which of the following measures of position is appropriate to use if we are looking for the
75% of the data?
A. quartile
B. percentile
C. decile
D. percentile rank
2. Which of the following measures of position is appropriate to use if we are looking for the
80% of the data?
A. quartile
B. percentile
C. decile
D. percentile rank
3. Which of the following measures of position is appropriate to use if we are looking for the
33% of the data?
A. quartile
B. percentile
C. decile
D. percentile rank
4. What percent of the data is above the 7 th decile?
A. 3%
B. 7%
C. 30%
26
D. 70%
The school principal of Malipayon High School surveyed the ages of his 40 teachers.
These are the data he collected:
36
42
31
37
32
25
31
51
24
24
46
50
50
37
44
23
43
42
35
49
45
52
44
39
30
46
31
46
27
39
42
45
5. What are the ages of teachers in the bottom 13%?
A. ≤ 13
B. ≤ 25
C. ≤ 30
D. ≤ 31
6. What are the ages of teachers in the bottom 30%?
A. ≤ 25
B. ≤ 30
C. ≤ 31
D. ≤ 70
7. What are the ages of teachers in the bottom 25%?
A. ≤ 25
B. ≤ 30
C. ≤ 31
D. ≤ 75
23
28
31
27
32
28
39
35
8. What measure of position is appropriate in finding the ages in the bottom 25%?
A. quartile
B. percentile
C. decile
D. percentile rank
9. What are the ages of teachers in the top 10%?
A. > 24
B. > 42
C. > 45
D. > 50
10. What are the ages of teachers in the top 25%?
A. > 30
B. > 42
C. > 45
D. > 50
11. What are the ages of teachers in the top 35%?
A. > 31
B. > 42
C. > 45
D. > 50
A teacher surveyed the distances of her students’ house from school in kilometres.
The data she collected from 35 students is presented in a frequency table below.
Class interval ( distances from school to
Frequency( No. of students)
students’ house in km)
12 – 14
5
9 – 11
3
6 –8
8
3–5
9
0–2
10
12. How far are the top 20% of the students’ house from school?
A. > 4.42 km
B. > 9.5 km
C. > 8.22 km
D. > 11.25 km
13. How far are the top 25% of the students’ house from school?
A. > 4.42 km
B. > 9.5 km
C. > 8.22 km
D. > 11.25 km
14. How far are the top 15% of the students’ house from school?
A. > 4.42 km
B. > 9.5 km
C. > 8.22 km
D. > 11.25 km
15. How near are the bottom 45% of the students’ house from school?
A. ≤ 4.42 km
B. ≤ 9.5 km
C. ≤ 8.22 km
D. ≤ 11.25 km
27
28
Lesson 2: What’s More
Lesson 2: What’s More
2.
a.
b.
c.
d.
1.
Grouped
Decile
Lowest 20%
Monthly Income
Frequency
(in Pesos)
e.
f.
g.
5
5,001 – 6,000
9
6,001 – 7,000
15
7,001 – 8,000
13
8,001 – 9,000
8
9,001 – 10,000
a. Ungrouped
b. Decile
c. Top 40%
d.
LB
Foods
Toron
Siomai
Banana Cue
Camote Cue
Puto
Sandwich
Bread
Hotdog
Cupcake
Pizza
<cf
5
5000.5
14
6000.5
29
7000.5
42
8000.5
50
9000.5
D2 Class= 10
D2 = 6556.056
The family income which is less than or equal
to Php6,556.06.
e.
f.
g.
Lesson 1: What’s More
Lesson 1: What’s More
2.
a.
b.
c.
d.
Profit
580
560
550
535
530
510
510
505
505
500
Position of D6= 6.6 or 7
D6= 535
Banana Cue, Siomai, and
Toron
What’s More
1.
Grouped
Quartile
Lowest 25%
a. Ungrouped
b. Quartile
c. Bottom 25%
d.
Monthly Income
Frequency
(in Pesos)
e.
f.
g.
Lesson 2
3
5,001 – 6,000
7
6,001 – 7,000
15
7,001 – 8,000
13
8,001 – 9,000
7
9,001 – 10,000
LB
Foods
Toron
Siomai
Banana Cue
Camote Cue
Puto
Sandwich
Bread
Hotdog
Cupcake
Pizza
<cf
3
5000.5
10
6000.5
25
7000.5
38
8000.5
45
9000.5
Q1 Class= 11.25
Q1 =7083.83
The family income which is less than or equal
to Php7,083.83.
e.
f.
g.
Profit
580
560
550
535
530
510
510
505
505
500
Position of Q1=2.75 or 3
Q1=505
Pizza, Cupcake, and
Hotdog
Lesson 1
What’s More
Key to Answers
29
Assessment
B
5.
C
4.
B
3.
C
2.
A
1.
15. A
10. C
14. D
D
9.
A
8.
B
7.
C
6.
11. B
12. B
What I Have Learned
25%
5.
1st quartile
4.
8.
50%
3.
7.
≤ Q1
2.
4
1.
6.
9.
14. 20th Percentile
2nd decile
13. 99%
90%
12. ≤ P68
≤ D3
11. 37%
20%
10. above
15. 15%
Lesson 3: What’s More
Lesson 3: What’s More
2.
a.
b.
c.
d.
1.
Grouped
Decile
Highest 55%
a. Ungrouped
b. Percentile
c. Bottom 15%
d.
Monthly Income
Frequency
(in Pesos)
e.
f.
g.
13. C
Assessment
What I Have Learned
5
5,001 – 6,000
9
6,001 – 7,000
15
7,001 – 8,000
13
8,001 – 9,000
8
9,001 – 10,000
LB
Foods
Toron
Siomai
Banana Cue
Camote Cue
Puto
Sandwich
Bread
Hotdog
Cupcake
Pizza
<cf
5
5000.5
14
6000.5
29
7000.5
42
8000.5
50
9000.5
P45 Class= 22.5
P45 = 7567.167
The family income which is greater than
Php7,567.17.
e.
f.
g.
Profit
580
560
550
535
530
510
510
505
505
500
Position of P15= 1.65 or 2
P15= 505
Pizza, Cupcake, and
Hotdog
Lesson 3
What’s More
References
Callanta, Melvin, Canonigo, Allan, Chua, Arnaldo, Cruz, Jerry, Esparrago, Mirla,
Garcia, Elino, Magnaye, Aries, Orines, Fernando, Perez, Rowena, and
Ternida, Concepcion. 2015. Mathematics 10 Learners Module. Philippines:
Rex Bookstore, Inc.
Dilao, Soledad, Orines, Fernando, and Bernabe, Julieta. 2009. Advance Algebra,
Trigonometry and Statistics. Quezon City: SD Publications, Inc.
Malaborbor, Pastor, Sabangan, Leticia, Carreon, Editha, and Lorenzo, Jose Ramon.
2003. Functions. Metro Manila: Diamond Offset Press, Inc.
Milefoot.com Mathematics. “Measures of
Position”.http://www.milefoot.com/math/stat/desc-positions.htm
Universal Class. “How to Determine Measures of Position”.
https://www.universalclass.com/articles/math/statistics/measures-of-positionpercentiles-quartiles.htm
For inquiries and feedback, please write or call:
Department of Education –Learning Resources Management and
Development Center(LRMDC)
DepEd Division of Bukidnon
Sumpong, Malaybalay City, Bukidnon
Telefax:
((08822)855-0048
E-mail Address:
bukidnon@deped.gov.ph
30