Download Math7-Q4-M10

Mathematics
7
Quarter 4
Self-Learning Module 10
Measures of Central Tendency
Module
10
of Ungrouped Data
Mathematics – Grade 7
Quarter 4 – Module 10: Measures of Central Tendency of Ungrouped Data
First Edition, 2020
Republic Act 8293, section 176 states that no copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the
government agency or office wherein the work is created shall be necessary for
exploitation of such work for profit. Such agency or office may, among other things,
impose as a condition the payment of royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand
names, trademarks, etc.) included in this module are owned by their respective
copyright holders. Every effort has been exerted to locate and seek permission to use
these materials from their respective copyright owners. The publisher and authors
do not represent nor claim ownership over them.
Published by the Department of Education – Schools Division of Pasig City
Development Team of the Self Learning Module
Writers: Ma. Flor A. Pornela
Editors: Ma. Victoria L. Peñalosa
Reviewers: Ma. Cynthia P. Badana
Illustrator: Name
Layout Artist: Name
Management Team:
Ma. Evalou Concepcion A. Agustin
OIC – Schools Division Superintendent
Carolina T. Rivera, CESE
OIC-Assistant Schools Division Superintendent
Manuel A. Laguerta, EdD
Chief, Curriculum Implementation Division
Victor M. Javeña EdD
Chief, School Governance and Operations Division
Education Program Supervisors
Librada L. Agon EdD (EPP/TLE/TVL/TVE)
Liza A. Alvarez (Science/STEM/SSP)
Bernard R. Balitao (AP/HUMSS)
Joselito E. Calios (English/SPFL/GAS)
Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports)
Wilma Q. Del Rosario (LRMS/ADM)
Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larangan)
Perlita M. Ignacio PhD (EsP)
Dulce O. Santos PhD (Kindergarten/MTB-MLE)
Teresita P. Tagulao EdD (Mathematics/ABM)
Printed in the Philippines by Department of Education – Schools Division of
Pasig City
Mathematics
Quarter 4
Module 10
Measures of Central Tendency
of Ungrouped Data
7
Introductory Message
For the Facilitator:
Welcome to the Mathematics Grade 7 Self – Learning Module on Measures of
Central Tendency of Ungrouped data!
This Self-learning module was collaboratively designed, developed and
reviewed by educators both from the Schools Division Office of Pasig City headed by
its Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A.
Agustin, in partnership with the City Government of Pasig through its mayor,
Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K
to 12 Curriculum using the Most Essential Learning Competencies (MELC) in
developing this instructional resource.
This learning material hopes to engage the learners into guided and
independent learning activities at their own pace and time. Further, this also aims
to help learners acquire the needed 21st century skills especially the 5 Cs, namely:
Communication, Collaboration, Creativity, Critical Thinking and Character while
taking into consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the
body of the module:
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.
As a facilitator you are expected to orient the learners on how to use this
module. You also need to keep track of the learners' progress while allowing them to
manage their own learning. Moreover, you are expected to encourage and assist the
learners as they do the tasks included in the module.
For the Learner:
Welcome to the Mathematics Grade Level 7 Self – Leaning Module on Measures
of central tendency of Ungrouped Data!
This self-learning module was designed to provide you with fun and
meaningful opportunities for guided and independent learning at your own pace and
time. You will be enabled to process the contents of the learning resource while being
an active learner.
This self-learning module has the following parts and corresponding icons:
Expectations - These are what you will be able to know after
completing the lessons in the module
Pretest - This will measure your prior knowledge and the
concepts to be mastered throughout the lesson.
Recap - This section will measure what learnings and skills
that you understand from the previous lesson.
Lesson- This section will discuss the topic for this module.
Activities - This is a set of activities you will perform.
Wrap Up- This section summarizes the concepts and
applications of the lessons.
Valuing-this part will check the integration of values in the
learning competency.
Posttest - This will measure how much you have learned from
the entire module.
EXPECTATION
1. To find the mean, median and mode of ungrouped data (M7SPIVf-g-1).
PRETEST
Directions: Read each item carefully. Choose the letter of the best answer.
1. What is the most stable and useful measure of central tendency?
A. mean
B. median
C. mode
D. range
2. When the data has extreme scores, what is the best measure of average?
A. mean
B. median
C. mode
D. range
For numbers 3 – 5,
The list gives the ages of students enrolled in swimming classes.
4
3
4
5
6
7
8
30
5
9
11
5
7
8
6
7
3. What is the median age of the students enrolled in swimming classes?
A. 6.5
B. 7
C. 7.5
D. 8
4. Which of the following describes the distribution?
A. unimodal
B. bimodal
C. trimodal
D. no mode
5. What is the mean age of the students?
A. 6.21
B. 7.34
C. 7.81
D. 8
RECAP
A. Find at least 3 words associated with measures of central tendency.
Words are arranged horizontally, vertically or diagonally.
M
E
D
I
A
N
E
F
I
S
D
S
E
W
R
R
D
I
E
A
Y
Q
T
E
D
O
H
D
A
T
A
Q
L
C
A
G
O
P
L
U
E
D
S
I
S
M
P
E
A
S
R
E
V
D
U
N
W
N
F
A
N
B
I
T
B. Identify the measure of central tendency that is most appropriate to
determine the following.
1.
2.
3.
4.
The
The
The
The
leading presidential candidate.
average annual salary of employees in a company.
average grade in mathematics of the students in class.
average rainfall for the month of August.
LESSON
A measure of central tendency is a single value that attempts to
describe a set of data by identifying the central position within that set of
data. As such, measures of central tendency are sometimes called measures
of central location. The mean (often called the average) is most likely the
measure of central tendency that you are most familiar with, but there are
others, such as the median and the mode.
Mean
The mean is the most popular among the measures of central tendency
for it is widely used. It is commonly called the average of a set of n numbers
and it is the sum of all numbers divided by n.
The formulas for the computation of the mean are as follows:
Formula
𝑥̅ =
∑𝑥
𝑁
where ∑𝑥 = the summation of x (sum of the
measures)
N = number of values of x.
Example 1 :
The number of hours each student spends in studying and doing
school projects each day is shown in the table.
Find the mean amount of time the students spend in studying and
doing school projects.
Kristel
4
Roger
3
Roland
3
Liza
4
April
5
Gabby
4
Kim
3
Solution:
𝑥̅ =
∑𝑥
𝑁
4+3+3+4+5+4+3
7
26
𝑥̅ =
7
𝑥̅ = 3.7
Thus, the mean or average time the students spend in studying and
doing school projects each day is 3.7 hours.
𝑥̅ =
Example 2 :
The set of data shows the age of 10 teachers of Pasig City Science High
School. Calculate the mean.
39
24
25
26
32
25
23
50
26
27
Solution:
𝑥̅ =
∑𝑥
𝑁
39 + 24 + 25 + 26 + 32 + 25 + 23 + 50 + 26 + 27
10
297
𝑥̅ =
10
𝑥̅ = 29. 70
Thus, the mean or the average of the ages is 29.70 or 30
𝑥̅ =
Weighted Mean
𝑥̅ =
∑𝑓𝑋
𝑁
where
f = frequency
X = score
∑𝑓𝑋 = sum of the product of frequency and
score
N = total frequency
To find the mean of a set of data in a frequency table,
1. Multiply each “X” in the set of data by “f” to obtain “fX”
2. Find the sum of “fX”
3. Find the sum of f, (N)
4. The mean, 𝑥̅ =
∑𝑓𝑋
𝑁
Example 3 :
Find the mean savings per student, given the frequency distribution
table.
Weekly Savings
(X)
Number of Students
(f)
Weekly Savings
Number of Students
(fX)
40
1
40
50
4
200
60
1
60
70
3
210
80
1
80
N = 10
∑𝑓𝑋 = 590
Solution:
𝑥̅ =
∑𝑓𝑋
𝑁
590
𝑥̅ =
10
𝑥̅ = 59
Therefore, the mean weekly savings per student is Php 59.00.
Median
The median is the middle score for a set of data that has been arranged
in order of magnitude (list after the scores are arranged in decreasing or
𝑁+1
increasing order). It is the value of the ( 2 )th item or position.
Example 1 :
The following are the number of students per section of Grade 7 level of
Rizal High School in Pasig City. Find the median.
55, 60, 48, 50, 52, 50 and 51.
Solution:
Arrange the numbers in ascending order
48, 50, 50, 51, 52, 55, 60
N=7
𝑁 + 1 th
)
2
7 + 1 th
𝑥̃ = (
𝑥̃ = (
2
)
score
score
8
𝑥̃ = (2)th score
𝑥̃ = 4th score
𝑥̃ = 51
Since 7 is an odd number, simply get the middle score, so the median is 51.
Example 2 :
The following set of data shows the height in centimeter of 10
students:
160, 154, 156, 160, 170, 159, 155, 145, 160 and 167.
Solution:
Arrange the numbers in ascending order
145, 154, 155, 156, 159, 160, 160, 160, 167, 170
N = 10
𝑁 + 1 th
)
2
10 + 1 th
𝑥̃ = (
𝑥̃ = (
2
11 th
𝑥̃ = ( 2 )
)
score
score
score
𝑥̃ = 5.5th score (the mean of the 5th and 6th scores)
Since the number of measures is even, then the median is the mean of
the two middle scores.
𝑥̃ =
159 + 160
2
𝑥̃ = 159.50
Hence, the median height of in centimeters of students is 159.50
Mode
The mode is the measure or value which occurs most frequently in a
set of data. It is the value with the greatest frequency. The word modal is
often used when referring to the mode of a data set. If a data set has only one
value that occurs most often, the set is called unimodal. A data set that has
two values that occur with the same greatest frequency is referred to
as bimodal. When a set of data has more than two values that occur with the
same greatest frequency, the set is called multimodal.
To find the mode for a set of data:
1. Select the measure that appears most often in the set.
2. If two or more measures appear the same number of times, then each of
these values is a mode.
3. If every measure appears the same number of times, then the set of data
has no mode.
Example 1 :
The sizes of 9 classes in a certain school are 58, 52, 58, 65, 68, 60, 57,64
and 60.
Solution:
𝑥̂ = 58 and 60 (Bimodal)
Example 2 :
The following data shows the height in centimeter of 10 students:
160, 154, 156, 160, 170, 159, 155, 145, 160 and 167.
Solution:
𝑥̂ = 160 (Unimodal)
ACTIVITIES
ACTIVITY 1: PRACTICE!
Direction: Calculate the mean of ungrouped data.
1. 10, 12, 15, 25
2. 16, 19, 20, 30, 10
3. 21, 29, 27, 33, 28
4. 28, 30, 32, 45, 40, 50
5. 65, 45, 75, 88, 45, 91
ACTIVITY 2: KEEP ON PRACTICING
Direction: Complete the table by calculating the median and mode.
Median
1.
2.
3.
4.
5.
Mode
3, 5, 2, 1, 2
5, 5, 2, 4, 6
8, 1, 4, 7, 7, 4
7, 6, 6, 8, 9, 10
11, 8, 9, 10, 10, 8, 9
ACTIVITY 3: TEST YOURSELF
Direction: Calculate the mean, median and mode of the following.
1. Shoe sizes of selected students: 5, 7, 6, 6.5, 8, 5, 5.5, 7, 7.5
2. Tickets sold in a week: 45, 35, 56, 44, 63, 55, 60
3. Expenses of Maria in 10 days: 100, 90, 125, 140, 130, 100, 125, 125,
100, 90
4. Weight of a grade 7 students in pounds: 75.2, 81.5, 70.4, 88.1, 90.7, 85.9
5. Height of basketball students: 156, 125, 136, 155, 169, 140, 145, 170,
181
WRAP UP
Remember…
Mean is the most popular among the measures of central tendency for
it is widely used. It is often called average.
The formulas for the computation of the mean are as follows.
Formula
𝑥̅ =
∑𝑥
Weighted Mean 𝑥̅ =
𝑁
∑𝑓𝑋
𝑁
Median is the middle score for a set of data that has been arranged in
order of magnitude (list after the scores are arranged in decreasing or
increasing order).
Mode is the number that occurs most frequently. If a data set has only
one value that occurs most often, the set is called unimodal. A data set that
has two values that occur with the same greatest frequency is referred to
as bimodal. When a set of data has more than two values that occur with the
same greatest frequency, the set is called multimodal.
To find the mode for a set of data:
1. Select the measure that appears most often in the set.
2. If two or more measures appear the same number of times, then each of
these values is a mode.
3. If every measure appears the same number of times, then the set of data
has no mode.
VALUING
Reflections: (Journal Writing)
As a student, why do you think your knowledge in finding the mean,
median and mode is important? Identify a real–life situation/s wherein you
can relate your knowledge about these measures of central tendency. Share
your thoughts and insights in 3-5 sentences and write these in your notebook.
POSTTEST
A. Directions: Read each item carefully. Choose the letter of the best
answer.
1. What is the measure of central tendency that divides an arranged
(ascending or descending) distribution into two equal parts?
A. mean
B. Median
C. Mode
D. Range
2. Which set of data has a mean of 15, a median of 14, and a mode of 14?
A. 3, 14, 14, 19, 25
B. 3, 7, 14, 15, 25
C. 4, 14, 14, 15, 22
D. 9, 14, 15, 15, 22
3. The mean score of a set of 40 scores is 81. Find the sum of the 40 test
scores.
A. 2, 250
B. 2, 280
C. 2, 820
D. 3, 240
B. Direction: Calculate for the mean, median and mode.
1. Mang Jose is a fish vendor. The following are his sales for a week.
Sunday: Php 7, 800
Monday: Php 3, 200
Tuesday: Php 4, 500
Wednesday: Php 3, 900
Thursday: Php 5, 100
Friday: Php 6,300
Saturday: Php 8,200
RECAP
A. Middle, Median, Data,
Frequent
B.
1. Mode
2. Mean
3. Mean
4. Mean
PRETEST
1.
2.
3.
4.
5.
A
C
A
B
C
POSTTEST
A.
1. B
2. A
3. D
ACTIVITY 1
1. 15.5
2. 19
3. 27.6
4. 37.5
5. 68.17
ACTIVITY 2
Median
2
5
5.5
7.5
9
1.
2.
3.
4.
5.
Mode
2
5
4 and 7
6
8, 9 and 10
ACTIVITY 3
Mean
Median
6.5
6.39
51.14 55
112. 5 112.5
81.97 83.7
153
155
1.
2.
3.
4.
5.
Mode
5, 7
no mode
100, 125
no mode
no mode
B.
Mean – 5, 571. 43
Median – 5, 100
Mode – no mode
KEY TO CORRECTION
REFERENCES
Oronce, Orlando, and Marilen Mendoza. E-MATH 7. Manila: Rex Book
Store, Inc., 2015.
Nivera, Gladys. Grade 7 Mathematics Patterns and Practicalities. Makati
City: Salesiana Books by Don Bosco Inc., 2014
Mirabona, Isaac, and Custodio, Sergio. Interactive Mathematics Grade 7.
Manila: Innovative Educational Materials Inc., 2013
http://www.riosalado.edu/web/oer/WRKDEV100
20011_INTER_0000_v1/lessons/Mod05_MeanMedianMode.shtml
(accessed July 18, 2020)
http://web.simmons.edu/~benoit/lis642/Hafner-Chapter5.pdf
(accessed July 18, 2020)
http://www.cimt.plymouth.ac.uk/mepjamaica/unit17/StudentText.pdf
(accessed July 18, 2020)